id string | prompt string | output_type string | domain string | evaluation_mode string | solvability int64 | numeric_value string | source_url string | source_note string | test_points list | metric_key string | optimization_direction string | baseline_value int64 | baseline_note string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
w4_watson_integral | Consider the following research problem in mathematics.
**Closed Form for the 4-Dimensional Lattice Green's Function ($W_4$)**
**Definition:** The Watson integrals $W_d$ represent the Green's function at the origin for the hypercubic lattice Green's function constant at the origin. They are defined by the integral: \[ W_d = \frac{1}{\pi^d} \int_0^{\pi} \cdots \int_0^{\pi} \frac{dx_1 \cdots dx_d}{d - \sum_{i=1}^d \cos x_i} \] While exact closed-form solutions exist for dimensions $d=1, 2, 3$ (involving $\Gamma$ functions and standard constants), the value for $d=4$, numerically approximated as $0.30986...\dots$, has no known symbolic expression.
**Task:** Find a symbolic closed-form expression for the 4-dimensional Watson integral $W_4$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 0 | 0.3098667804621204281696744162147501775383222672904396642383504626790703346638908327580983261838473482149795083 | https://arxiv.org/pdf/1801.02182 | Zhou, 'On Laporta's 4-loop sunrise formulae' (2018) - Laporta (2018) conjectures a closed-form, and Zhou proves it, giving a hypergeometric/Gamma expression | null | null | null | null | null |
w5_watson_integral | Consider the following research problem in mathematics.
**Closed Form for the 5-Dimensional Lattice Green's Function ($W_5$)**
**Definition:** The Watson integrals $W_d$ represent the Green's function at the origin for the hypercubic lattice Green's function constant at the origin. They are defined by the integral: \[ W_d = \frac{1}{\pi^d} \int_0^{\pi} \cdots \int_0^{\pi} \frac{dx_1 \cdots dx_d}{d - \sum_{i=1}^d \cos x_i} \] For dimension $d=5$, the numerical value is approximately $0.23126...\dots$. Despite the existence of solutions for lower dimensions, no closed-form expression using standard mathematical constants and special functions is known for $W_5$.
**Task:** Find a symbolic closed-form expression for the 5-dimensional Watson integral $W_5$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.23126162496804623574142702438771339710908546970102847765391320224201754069413746234473308609901834330534861291 | https://arxiv.org/abs/1004.1435 | Guttmann, 'Lattice Green functions in all dimensions' (2010) - covers Watson integrals W_d for arbitrary d-dimensional hypercubic lattices | null | null | null | null | null |
w6_watson_integral | Consider the following research problem in mathematics.
**Closed Form for the 6-Dimensional Lattice Green's Function ($W_6$)**
**Definition:** The Watson integrals $W_d$ represent the Green's function at the origin for the hypercubic lattice Green's function constant at the origin. They are defined by the integral: \[ W_d = \frac{1}{\pi^d} \int_0^{\pi} \cdots \int_0^{\pi} \frac{dx_1 \cdots dx_d}{d - \sum_{i=1}^d \cos x_i} \] For dimension $d=6$, the numerical value is approximately $0.18616...\dots$. This problem likely requires cross-domain synthesis, as $W_6$ may involve more complex structures than the Gamma function products found in $W_3$.
**Task:** Find a symbolic closed-form expression for the 6-dimensional Watson integral $W_6$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.18616056220444530728094072199476887544269877039883875411399992156674267940911681325387509047530591295459637041 | https://arxiv.org/abs/1004.1435 | Guttmann, 'Lattice Green functions in all dimensions' (2010) - comprehensive treatment of lattice Green functions and Watson integrals in all dimensions | null | null | null | null | null |
bessel_moment_c5_0 | Consider the following research problem in mathematics.
**Closed Form for the Bessel Moment $c_{5,0}$**
**Definition:** The Bessel function moments are defined by the integral $c_{n,k} = \int_0^{\infty} t^k K_0(t)^n \, dt$, which arise in $(n-1)$-loop Feynman diagram calculations. For $n=5, k=0$, the value is approximately $135.26...\dots$. Here $c_{n,k}$ means exactly $\int_0^\infty t^k K_\theta(t)^n \, dt$ (no additional $t \, dt$ factor), and $K_0$ is the modified Bessel function of the second kind, order 0, on $(0, \inf)$. The closed form for $c_{5,0}$ is unknown.
**Task:** Find a symbolic closed-form expression for the Bessel moment $c_{5,0}$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 135.26830258086883759422627964619220742030588935942352678469351371045888711773849131554701138246193550710196669 | https://arxiv.org/abs/0801.0891 | Bailey, Borwein, Broadhurst, Glasser, 'Elliptic integral evaluations of Bessel moments' (2008) - closed forms for c_{n,k} Bessel moments with progress on c_{5,0} | null | null | null | null | null |
bessel_moment_c6_0 | Consider the following research problem in mathematics.
**Closed Form for the Bessel Moment $c_{6,0}$**
**Definition:** The Bessel function moments are defined by $c_{n,k} = \int_0^{\infty} t^k K_0(t)^n \, dt$. For the case $n=6, k=0$, the numerical value is approximately $809.62...\dots$. Here $c_{n,k}$ means exactly $\int_0^\infty t^k K_\theta(t)^n \, dt$ (no additional $t \, dt$ factor), and $K_0$ is the modified Bessel function of the second kind, order 0, on $(0, \inf)$. No closed-form solution has been discovered.
**Task:** Find a symbolic closed-form expression for the Bessel moment $c_{6,0}$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 809.62084822486627594007354000392747913008434556749563772879133821833933609599367021661064055934872732418948686 | https://arxiv.org/abs/0801.0891 | Bailey, Borwein, Broadhurst, Glasser, 'Elliptic integral evaluations of Bessel moments' (2008) - formulae for integrals of products of six or fewer Bessel functions | null | null | null | null | null |
bessel_moment_c5_1 | Consider the following research problem in mathematics.
**Closed Form for the Bessel Moment $c_{5,1}$**
**Definition:** The Bessel function moments are defined by $c_{n,k} = \int_0^{\infty} t^k K_0(t)^n \, dt$. This problem concerns the first moment ($k=1$) with $n=5$ Bessel functions. The numerical value is approximately $2.4965...\dots$. Here $c_{n,k}$ means exactly $\int_0^\infty t^k K_\theta(t)^n \, dt$ (no additional $t \, dt$ factor), and $K_0$ is the modified Bessel function of the second kind, order 0, on $(0, \inf)$.
**Task:** Find a symbolic closed-form expression for the Bessel moment $c_{5,1}$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 2.4965992507497653561840017811514997432406114327981162232729101382421014141270463045039463065513848490719149810 | https://arxiv.org/abs/0801.0891 | Bailey, Borwein, Broadhurst, Glasser, 'Elliptic integral evaluations of Bessel moments' (2008) - substantial progress on c_{5,2k+1} odd moments | null | null | null | null | null |
box_integral_b6_1 | Consider the following research problem in mathematics.
**Closed Form for the 6D Box Integral $B_6(1)$**
**Definition:** The box integral $B_n(s)$ measures the $s$-th moment of the Euclidean distance from the origin to a point in the unit hypercube $[0,1]^n$: \[ B_n(s) = \int_{[0,1]^n} |\mathbf{x}|^s \, d\mathbf{x} \] For $n=6$ and $s=1$, the value is approximately $1.3885...\dots$. Closed forms exist for $n \le 5$, but $B_6(1)$ remains open.
**Task:** Find a symbolic closed-form expression for the box integral $B_6(1)$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 1.388574084457347842530254073030788815910945088782207029758933139762637896937682885791843577 | https://www.osti.gov/biblio/964379 | Bailey, Borwein, Crandall, 'Higher-dimensional box integrals' (2010) - first nontrivial closed forms for six-dimensional box integrals and also provides closed forms for $n$ up to 5. | null | null | null | null | null |
box_integral_b7_1 | Consider the following research problem in mathematics.
**Closed Form for the 7D Box Integral $B_7(1)$**
**Definition:** The box integral $B_n(s)$ measures the $s$-th moment of the Euclidean distance from the origin to a point in the unit hypercube $[0,1]^n$: \[ B_n(s) = \int_{[0,1]^n} |\mathbf{x}|^s \, d\mathbf{x} \] For $n=7$ and $s=1$ has a numerical value of approximately $2.1031...\dots$. No closed-form expression is currently known.
**Task:** Find a symbolic closed-form expression for the box integral $B_7(1)$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 2.1031677468737035517164242261635051336191256398255234438587726962237281589021474209489946038383277181415894854 | https://www.osti.gov/biblio/964379 | Bailey, Borwein, Crandall, 'Higher-dimensional box integrals' (2010) - first nontrivial closed forms for six-dimensional box integrals and also provides closed forms for $n$ up to 5. | null | null | null | null | null |
box_integral_b5_neg2 | Consider the following research problem in mathematics.
**Closed Form for the Box Integral $B_5(-2)$**
**Definition:** The box integral $B_n(s) = \int_{[0,1]^n} |\mathbf{x}|^s \, d\mathbf{x}$ generally becomes harder for negative $s$. For $n=5$ and $s=-2$, the value is approximately $0.76560...\dots$. This represents an expectation of the inverse squared distance in 5 dimensions.
**Task:** Find a symbolic closed-form expression for the box integral $B_5(-2)$ with Euclidean norm.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 0 | 0.76560088060035042048313592041746790597916235131578395215189528953020852443035092982996181509585989486734309034 | https://www.osti.gov/biblio/964379 | Bailey, Borwein, Crandall, 'Higher-dimensional box integrals' (2010) - first nontrivial closed forms for six-dimensional box integrals and also provides closed forms for $n$ up to 5. | null | null | null | null | null |
elliptic_k_moment_3 | Consider the following research problem in mathematics.
**Third Moment of the Complete Elliptic Integral $K(k)$**
**Definition:** This problem asks for the closed form of the moment integral $\int_0^1 K(k)^3 \, dk$, where $K(k)$ is the complete elliptic integral of the first kind. The numerical value is approximately $7.0902...\dots$. While the first and second moments are known (involving Catalan's constant and $\zeta(3)$), the third moment is open.
**Task:** Find a symbolic closed-form expression for $\int_0^1 K(k)^3 \, dk$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 0 | 7.0902270048462694609898023700595492524524185476584179865587158041145846347861787736244562389891764350266529514 | https://arxiv.org/abs/1303.2259 | Rogers, Wan, Zucker: 'Moments of elliptic integrals and critical L-values'. Ramanujan J. 37 (2015), 113-130. Provides a closed form for the third moment of K(k) expressible via gamma functions | null | null | null | null | null |
elliptic_k_moment_4 | Consider the following research problem in mathematics.
**Fourth Moment of the Complete Elliptic Integral $K(k)$**
**Definition:** This problem asks for the closed form of the moment integral $\int_0^1 K(k)^4 \, dk$, where $K(k)$ is the complete elliptic integral of the first kind and $K(k)=\int_{0}^{\pi/2} \frac{d heta}{\sqrt{1-k^2 \sin^2( heta)}}$. The numerical value is approximately $15.611...\dots$.
**Task:** Find a symbolic closed-form expression for $\int_0^1 K(k)^4 \, dk$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 15.611523683715693929074704703647595914409260699418022257962398941624312278709557178035465062471152754769332293 | https://arxiv.org/abs/1303.2259 | Rogers, Wan, Zucker: 'Moments of elliptic integrals and critical L-values'. Ramanujan J. 37 (2015), 113-130. Derives closed forms for elliptic integral moments expressible via gamma functions | null | null | null | null | null |
elliptic_k2_e_moment | Consider the following research problem in mathematics.
**Mixed Moment of Elliptic Integrals $K(k)^2 E(k)$**
**Definition:** This problem concerns the integral of the product of the square of the complete elliptic integral of the first kind $K(k)$ and the complete elliptic integral of the second kind $E(k)$: $\int_0^1 K(k)^2 E(k) \, dk$. The numerical value is approximately $4.7268...\dots$.
**Task:** Find a symbolic closed-form expression for $\int_0^1 K(k)^2 E(k) \, dk$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 0 | 4.7268180032308463073265349133730328349682790317722786577058105360763897565241191824163041593261176233511019676 | https://arxiv.org/abs/0801.0891 | Wan: 'Moments of products of elliptic integrals'. (2018). Develops closed forms for Bessel moments with connections to elliptic integrals | null | null | null | null | null |
airy_moment_a4 | Consider the following research problem in mathematics.
**Fourth Moment of the Airy Function ($a_4$)**
**Definition:** The Airy power moments are defined by $a_n = \int_0^\infty \mathrm{Ai}(x)^n \, dx$. These moments appear in random matrix theory. The fourth moment $a_4$ has the numerical value approx.\ $0.0046380...\dots$.
**Task:** Find a symbolic closed-form expression for the Airy moment $a_4$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 0 | 0.0046380290604946057287443641210015069017195022230366911564643170644289766133364996131025023047197563677273764507 | https://dlmf.nist.gov/9.11 | DLMF Section 9.11: Products of Airy Functions. The closed form is ln(3)/(24*pi^2) | null | null | null | null | null |
airy_moment_a5 | Consider the following research problem in mathematics.
**Fifth Moment of the Airy Function ($a_5$)**
**Definition:** The Airy power moments are defined by $a_n = \int_0^\infty \mathrm{Ai}(x)^n \, dx$. For $n=5$, the value is approximately $0.0013493...\dots$.
**Task:** Find a symbolic closed-form expression for the Airy moment $a_5$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | integrals | ground_truth_computable | 2 | 0.0013493589835177305394535748997338260553653997404797424839336973256901140935986288565766973541821804238164374932 | https://link.springer.com/article/10.1007/BF00942815 | Laurenzi, B.J. 'Moment integrals of powers of airy functions.' Z. angew. Math. Phys. 44, 891-908 (1993. Studies powers of the Airy function Ai(z) and its derivative Ai'(z). | null | null | null | null | null |
central_binomial_s5 | Consider the following research problem in mathematics.
**Closed Form for Central Binomial Sum $S_5$**
**Definition:** The series is defined as $S_k = \sum_{n=1}^\infty \frac{1}{n^k \binom{2n}{n}}$. Known results exist for $k=1, 2, 3, 4$ involving $\pi$, Clausen functions, and polylogarithms. The case $k=5$ (approx.\ $0.50542...$) is unsolved.
**Task:** Find a symbolic closed-form expression for the series sum $S_5$.
Even if your solution is unproven, you must propose an accurate candidate solution (such as a conjecture) that satisfies the requirements listed above and adequately addresses the task. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 0 | 0.50542947468351924164245048190843214918866901456826286498266471287573347337617590682716453318150013661960285541 | https://arxiv.org/abs/hep-th/0004153 | Borwein, Broadhurst, Kamnitzer: 'Central Binomial Sums, Multiple Clausen Values and Zeta Values', Exper. Math. 10 (2001), 25-34. Finds relationships between zeta values and central binomial sums | null | null | null | null | null |
autocorr_upper | Consider the following optimization problem.
**Improve Upper Bound on Autocorrelation Constant $C$**
**Definition:** The autocorrelation constant $C$ is defined as $C = \inf_f \frac{\max_{t} (f * f)(t)}{(\int f(x)\, dx)^2}$ where the infimum is over all non-negative, not identically zero functions $f$ supported on $[-1/4, 1/4]$, and $(f * f)(t) = \int f(t-x) f(x)\, dx$ is the autoconvolution. The current best bounds are $1.28 \leq C \leq 1.5028...$. The upper bound is from Yuksekgonul et al. (2025), and the lower bound is from Cloninger & Steinerberger (2017).
**Task:** Construct an explicit non-negative function $f$ supported on $[-1/4, 1/4]$ that achieves $\max_t (f*f)(t) / (\int f)^2 < 1.5028...$, improving the best known upper bound on $C$.
**Current State-of-the-Art:**
- Metric: Autoconvolution Ratio Upper Bound
- Best Known Value: 1.5028...
- Direction: MINIMIZE (lower is better)
- Source: Learning to Discover at Test Time (2025) by Mert Yuksekgonul, Daniel Koceja, Xinhao Li, Federico Bianchi, Jed McCaleb, Xiaolong Wang, Jan Kautz, Yejin Choi, James Zou, Carlos Guestrin, and Yu Sun
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Return non-negative step heights for N equal-width subintervals of [-1/4, 1/4].
# The validator computes: max_t (f*f)(t) / (integral f)^2 for the step function.
return {"values": [v_0, v_1, ..., v_N_minus_1]}
# or simply: return [v_0, v_1, ..., v_N_minus_1]
```
**Constraints:**
- Values must be non-negative finite real numbers
- At least 100 intervals required
**To beat the baseline:** Your result must be < 1.5028...
| construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2601.16175 | Yuksekgonul et al. (2025) 'Learning to Discover at Test Time' (arXiv:2601.16175) - achieves C ≤ 1.50286 via a 30000-piece step function, and Cloninger & Steinerberger (2017) 'On Suprema of Autoconvolutions with an Application to Sidon sets' (Proc. AMS 145(8):3191-3200, arXiv:1403.7988) | null | null | null | null | null |
autocorr_signed_upper | Consider the following optimization problem.
**Signed Autocorrelation Constant $C'$ Upper Bound**
**Definition:** The signed autocorrelation constant $C'$ is defined as $C' = \inf_f \max_t (f * f)(t) / (\int f)^2$, where the infimum is over all not identically zero functions $f$ (which may take negative values) supported on $[-1/4, 1/4]$, and $(f * f)(t) = \int f(t-x) f(x)\, dx$ is the autoconvolution. Unlike the unsigned version, $f$ is not restricted to be non-negative. The current best upper bound is $C' \leq 1.4557$.
**Task:** Construct an explicit step function $f$ supported on $[-1/4, 1/4]$ that achieves $\max_t (f*f)(t) / (\int f)^2 < 1.4557$, improving the best known upper bound on $C'$.
**Current State-of-the-Art:**
- Metric: Signed Autocorrelation Constant C' Upper Bound
- Best Known Value: 1.4557
- Direction: MINIMIZE (lower is better)
- Source: AlphaEvolve: A coding agent for scientific and algorithmic discovery (2025) by Alexander Novikov et al.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Return a list of real-valued step heights for N equal-width
# subintervals of [-1/4, 1/4]. Values may be positive or negative.
# The validator computes: 2N * max(convolve(a, a)) / (sum(a))^2
# which equals max_t (f*f)(t) / (integral f)^2 for the step function.
return {"values": [v_0, v_1, ..., v_N_minus_1]}
# or simply: return [v_0, v_1, ..., v_N_minus_1]
**Constraints:**
- Values must be finite real numbers (no NaN or inf)
- sum(values) must be nonzero
- At least 10 intervals required
**To beat the baseline:** Your result must be < 1.4557
| construction | combinatorics | benchmark_best_known | 2 | null | https://arxiv.org/abs/1205.0626 | Jedwab, Katz & Schmidt (2013) 'Advances in the merit factor problem for binary sequences' - establishes asymptotic merit factor bounds and addresses signed autocorrelation properties | null | null | null | null | null |
resultant_chebyshev | Consider the following research problem in mathematics.
**Resultant of Chebyshev and Legendre Polynomials**
**Definition:** Let $T_n(x) = \cos(n \arccos x)$ be the Chebyshev polynomial of the first kind of degree $n$, and let $P_m(x)$ be the Legendre polynomial of degree $m$, defined by $(1 - 2xt + t^2)^{-1/2} = \sum_{m=0}^\infty P_m(x) t^m$. Consider the polynomial resultant $\mathrm{Res}_x(T_n, P_m)$ for specific $(n, m) = (30, 20)$.
**Task:** Find a symbolic closed-form formula for $\mathrm{Res}_x(T_{30}, P_{20})$, the resultant of the Chebyshev polynomial $T_{30}(x)$ and the Legendre polynomial $P_{20}(x)$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 0 | 3.50250188617129022035975427961480421661370306852776070285584178979291528698154779416561876786842808192139e+146 | https://en.wikipedia.org/wiki/Chebyshev_polynomials | Resultant Res_x(T_30, P_20) of Chebyshev T_30 and Legendre P_20 polynomials. While Res(T_n, T_m) and Res(T_n, U_m) have known closed forms (Gishe-Ismail 2008), no general closed-form formula is known for cross-family Res(T_n, P_m). | null | null | null | null | null |
quartic_oscillator_lambda | Consider the following open problem in mathematical physics.
**Eigenvalues of a Quartic Oscillator with Quadratic Parameter**
**Definition:** In units where \(\hbar=m=1\), define \(\varepsilon_n(\lambda)\) as the \(n\)-th eigenvalue of
\[ -\tfrac12\,\psi''(x) + \Big(\tfrac{x^4}{4} - \tfrac{\lambda x^2}{2}\Big)\psi(x) = \varepsilon\,\psi(x),\quad \psi(x)\to 0\ \text{as }|x|\to\infty. \]
**Task:** Propose a *closed-form* formula for \(\varepsilon_n(\lambda)\) that matches the provided high-precision test points.
**Hard restrictions (enforced by a policy checker after numeric pass):**
- Do NOT perform numerical eigenvalue computations or diagonalization (e.g. no `mp.eig*`, no matrix truncation + `eigh`, no shooting methods).
- Do NOT use numerical root finding (e.g. no `findroot`), numerical quadrature (e.g. no `quad`), or truncated approximation loops / `nsum` / `nprod`.
- Use only explicit expressions built from mpmath constants, elementary functions, and named special functions.
REQUIRED OUTPUT FORMAT:
def proposed_solution(n, lam):
from mpmath import mp
mp.dps = 100
# closed-form expression only
result = ...
return result
| function | continuum_physics | ground_truth_computable | 2 | null | https://dft.uci.edu/pubs/OB20.pdf | Problem definition (Schr\u00f6dinger equation and potential v_\u03bb) follows Okun & Burke (2020). Published 40-digit eigenvalue benchmarks are in the Supplemental Information Table S1: https://dft.uci.edu/pubs/OB20s.pdf. The paper explicitly notes the quartic oscillator lacks a simple analytic solution, supporting the 'unknown simple closed form' premise. | [
{
"args": [
0,
0
],
"expected": "0.420804974475447763207338706947"
},
{
"args": [
1,
0
],
"expected": "1.50790124116048221411837172711"
},
{
"args": [
0,
1
],
"expected": "0.147235140090035649969124897756"
},
{
"args": [
2... | null | null | null | null |
spheroidal_eigenvalue_lambda_m0 | Consider the following open problem in spectral theory / special functions.
**Angular Prolate Spheroidal Eigenvalues (order m = 0)**
Let \(c \ge 0\) be a real parameter. Consider the Sturm-Liouville eigenvalue problem on \((-1,1)\):
\[
-\frac{d}{dx}\Big((1-x^2)\,y'(x)\Big) + c^2 x^2\,y(x) = \lambda\,y(x),\qquad -1<x<1,
\]
with the boundary condition that \(y(x)\) remains bounded as \(x\to\pm 1\).
It is known that the spectrum is discrete and real:
\(0\le \lambda_0(c) < \lambda_1(c) < \lambda_2(c) < \cdots\), and \(\lambda_n(0)=n(n+1)\).
**Task:** Find a symbolic closed-form expression for the eigenvalue function \(\lambda_n(c)\) valid for general integer \(n\ge 0\) and real \(c\ge 0\).
Your expression must be a *finite* combination of:
- Rational and algebraic numbers
- Constants: \(\pi, e, \gamma, G, \zeta(k)\)
- Named special functions available in mpmath (e.g. \(\Gamma\), polylogarithms, hypergeometric functions, elliptic integrals at explicit arguments, etc.)
INADMISSIBLE (will be rejected by compliance checks):
- Any numerical eigenvalue computation or diagonalization (including mp.eig, mp.eigsy, mp.eigh, mp.matrix, or any linear-algebra-based spectral routines)
- Numerical root-finding (findroot), numerical quadrature (quad), ODE/PDE solvers, continued-fraction solvers, or any iterative/truncation loops intended to approximate \(\lambda_n(c)\)
- Unevaluated integrals or implicit definitions (e.g. “\(\lambda\) is the root of …”)
- Hardcoding a lookup table for specific test points
REQUIRED OUTPUT FORMAT:
Write a Python function using only mpmath that evaluates your proposed closed form:
```python
def proposed_solution(n, c):
from mpmath import mp
mp.dps = 100
# Use only explicit closed-form combinations of mpmath constants/functions.
# No numerical spectral methods, no root-finding, no quadrature, no approximation loops.
result = ...
return result
```
| function | continuum_physics | ground_truth_computable | 2 | null | https://arxiv.org/abs/math-ph/0212051 | Falloon, Abbott, Wang (2003). Journal of Physics A: Mathematical and General. 'Theory and computation of spheroidal wavefunctions.' Background: spheroidal eigenvalues are typically computed numerically (continued fractions / tridiagonal-matrix truncations) and only limited analytic identities/special cases are available. Note: we use the Flammer convention for the operator, from C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, 1957). If using Falloon et al.'s Meixner notation, add c^2 to their eigenvalue---see Appendix B for their conversion, \lambda_{n,m}^{Flammer}(c)=\lambda_{n,m}^{Meixner}(c)+c^2. | [
{
"args": [
0,
0.5
],
"expected": "0.0824148067425269924096946397534948364330689601763843733394727641071589362064113744154336157"
},
{
"args": [
1,
0.5
],
"expected": "2.14957048639105664049176196387542651912690409940545857071509879213413280098798136719898456"
}... | null | null | null | null |
feigenbaum_delta | Consider the following research problem in mathematics.
**Closed Form for the Feigenbaum Constant $\delta$**
**Definition:** The Feigenbaum constant $\delta$ is the limiting ratio of consecutive bifurcation intervals in the period-doubling route to chaos for unimodal maps. For the logistic map $f(x) = rx(1-x)$, if $r_n$ denotes the parameter value at which a $2^n$-cycle appears, then $\delta = \lim_{n \to \infty} (r_{n-1} - r_{n-2})/(r_n - r_{n-1}) = 4.6692...$. This constant is universal across all maps with quadratic maxima. It is not even proven to be irrational, though it is believed to be transcendental.
**Task:** Find a symbolic closed-form expression for the Feigenbaum constant $\delta = 4.6692...$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 3 | 4.6692016091029906718532038204662016172581855774757686327456513430041343302113147371386897440239480138171659848 | https://oeis.org/A006890 | OEIS decimal expansion of Feigenbaum bifurcation velocity constant delta = 4.669201609102990671853...; no closed form known | null | null | null | null | null |
feigenbaum_alpha | Consider the following research problem in mathematics.
**Closed Form for the Feigenbaum Constant $\alpha$**
**Definition:** The Feigenbaum constant $\alpha$ governs the geometric scaling of the attractor in period-doubling bifurcations. It is defined as the limit $\alpha = \lim_{n \to \infty} d_n / d_{n+1}$ (quadratic-unimodal fixed-point normalization with $\alpha > 0$), where $d_n$ is the width of the central region of the $2^n$-cycle attractor. The value is approximately $\alpha = 2.5029...$. It is in the quadratic unimodal universality class.
**Task:** Find a symbolic closed-form expression for the Feigenbaum constant $\alpha = 2.5029...$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 3 | 2.50290787509589282228390287321821578638127137672714997733619205677923546317959020670329964974643383412959 | https://oeis.org/A006891 | OEIS decimal expansion of Feigenbaum reduction parameter alpha = 2.502907875095892822283...; no closed form known | null | null | null | null | null |
fransen_robinson_constant | Consider the following research problem in mathematics.
**Closed Form for the Fransén-Robinson Constant**
**Definition:** The Fransén-Robinson constant $F$ is defined by the integral $F = \int_0^{\infty} \frac{1}{\Gamma(x)}\,dx$, where $\Gamma$ is the Euler gamma function. Its numerical value begins $2.8077...\dots$ (OEIS A058655). The constant arises in the study of the reciprocal gamma function and its integral representations. Despite extensive numerical computation, no closed-form expression for $F$ in terms of standard mathematical constants and special functions is known.
**Task:** Find a symbolic closed-form expression for the Fransén-Robinson constant $F = \int_0^{\infty} \frac{1}{\Gamma(x)}\,dx$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 2 | 2.8077702420285193652215011865577729323080859209301982912200548095971008891219016655101853081681966381418741643 | https://oeis.org/A058655 | OEIS A058655: Decimal expansion of the Fransén-Robinson constant; no closed form known | null | null | null | null | null |
nested_radical_kasner | Consider the following research problem in mathematics.
**Closed Form for the Nested Radical Constant**
**Definition:** The nested radical constant (also called Kasner's number) is defined as the limit of the nested radical expression $\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}$. Its numerical value begins $1.7579...\dots$ (OEIS A072449). The constant arises from Edward Kasner's work on nested radicals. Despite its simple definition, no closed-form expression in terms of known mathematical constants or special functions has been found.
**Task:** Find a symbolic closed-form expression for Kasner's nested radical constant $\sqrt{1 + \sqrt{2 + \sqrt{3 + \cdots}}}$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 2 | 1.7579327566180045327088196382181385276531999221468377043101355003851102326744467575723445540002594529709324718 | https://oeis.org/A072449 | OEIS A072449: Decimal expansion of Kasner's number sqrt(1+sqrt(2+sqrt(3+...))); no closed form known. Herschfeld (1935) in 'On Infinite Radicals' says Kasner suggested investigation of “infinite radicals” and introduces K as the 'Kasner number.' | null | null | null | null | null |
mrb_constant | Consider the following research problem in mathematics.
**Closed Form for the MRB Constant**
**Definition:** The MRB constant (named after Marvin Ray Burns) is defined as the alternating sum $M = \sum_{n=1}^{\infty} (-1)^n (n^{1/n} - 1)$. Its numerical value begins $0.18785...\dots$. The constant arises in the study of the asymptotic behavior of $n$-th roots. Despite extensive computation to millions of digits, no closed-form expression for $M$ in terms of standard mathematical constants and special functions is known.
**Task:** Find a symbolic closed-form expression for the MRB constant $M = \sum_{n=1}^{\infty} (-1)^n (n^{1/n} - 1)$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 2 | 0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443310749690384234586 | https://oeis.org/A037077 | OEIS A037077: Decimal expansion of the MRB constant sum((-1)^n*(n^(1/n)-1)); no closed form known. There are known forms that are not closed-form, such as an infinite series involving derivatives of the Dirichlet eta function and an integral representation according to MathWorld's article, 'https://mathworld.wolfram.com/MRBConstant.html'. | null | null | null | null | null |
torsional_rigidity_square | Consider the following research problem in mathematics.
**Closed Form for the Torsional Rigidity Ratio of a Square**
**Definition:** The torsional rigidity of a prismatic bar with a full side length $b$ is characterized by the dimensionless ratio $J/b^4$, where $J$ is the torsion constant. Using Saint-Venant's classical solution, this ratio is given by the series $J/b^4 = \frac{1}{3}\left[1 - \frac{192}{\pi^5}\sum_{n=0}^{\infty} \frac{\tanh((2n+1)\pi/2)}{(2n+1)^5}\right]$. Its numerical value begins $0.14057...\dots$. Despite the explicit series representation, it is unknown whether this constant can be expressed in closed form using standard mathematical constants and special functions without infinite summation.
**Task:** Find a symbolic closed-form expression for the torsional rigidity ratio $J/b^4$ of a square cross-section.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | 0.140577014955153715588468730737731115267593118830092268073958148912875912876 | https://oeis.org/A180309 | OEIS entry for the decimal expansion of the torsional rigidity constant for a square shaft. MathWorld confirms the numerical value, 'Torsional Rigidity'. | null | null | null | null | null |
bernstein_constant | Consider the following research problem in mathematics.
**Closed Form for Bernstein's Constant**
**Definition:** Let $P^*_n$ denote the polynomial of degree $\le n$ that minimizes $\sup_{x \in [-1,1]} ||x| - P^*_n(x)|$. Define $E_n = \sup_{x \in [-1,1]} ||x| - P^*_n(x)|$. Bernstein's constant is $\beta = \lim_{n \to \infty} 2n \cdot E_{2n}$, where the limit uses even-degree polynomials only (since they respect the symmetry of $|x|$). Bernstein conjectured $\beta = 1/(2\sqrt{\pi}) \approx 0.28209\ldots$ in 1914, but this was disproved by Varga \& Carpenter (1987) who showed the values differ at the 3rd decimal place. No other conjectured closed form exists.
**Task:** Find a symbolic closed-form expression for Bernstein's constant $\beta$. Note: the conjecture $\beta = 1/(2\sqrt{\pi})$ has been disproved and is not an acceptable answer.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | mathematical_constants | ground_truth_computable | 2 | 0.28016949902386913303643649123067200004248213981236 | https://oeis.org/A073001 | Varga & Carpenter, Constr. Approx. 1 (1985) 333-348; Lubinsky (2003) integral representation | null | null | null | null | null |
townes_soliton | Consider the following research problem in mathematics.
**Townes Soliton Critical Mass (2D Cubic NLS Ground State Norm)**
**Definition:** Let $Q(r)$ be the unique positive radial solution of the ODE $Q''(r) + (1/r)Q'(r) - Q(r) + Q(r)^3 = 0$ for $r > 0$, with $Q'(0) = 0$ and $Q(r) \to 0$ as $r \to \infty$ (uniqueness: Kwong, Arch. Rational Mech. Anal. 105, 1989). This is the radial reduction of $\Delta Q - Q + Q^3 = 0$ in $\mathbb{R}^2$. The Townes soliton critical mass is $N_c = 2\pi \int_0^{\infty} Q(r)^2 \, r \, dr$ (the squared $L^2$ norm of $Q$ on $\mathbb{R}^2$). This constant determines the sharp constant in the 2D Gagliardo-Nirenberg inequality (Weinstein, Comm. Math. Phys. 87, 1983). No closed-form expression is known despite $N_c$ being central to the blow-up theory of the 2D focusing cubic NLS.
**Task:** Find a symbolic closed-form expression for the Townes soliton critical mass $N_c$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | 11.70089652455965387865397 | https://math.unm.edu/~plushnik/publications/LushnikovVladimirovaOptLett2014.pdf | Lushnikov and Vladimirova (2014). Optics Letters, v.39, 3429-3432, 'Nonlinear combining of laser beams.' They define the Townes soliton and provide N_c up to 1.7008965... | null | null | null | null | null |
mahler_1_x_y_z_w | Consider the following research problem in mathematics.
**Mahler Measure of $1+x+y+z+w$**
**Definition:** The logarithmic Mahler measure of the 4-variable polynomial $P(x,y,z,w) = 1+x+y+z+w$ is defined by the integral over the unit torus, and $m(P) = \int_0^1 \cdots \int_0^1 \log |P(e^{2\pi i t_1}, \dots, e^{2\pi i t_n})| \, dt_1 \cdots dt_n = \frac{1}{(2\pi)^n} \int_0^{2\pi} \cdots \int_0^{2\pi} \log |P(e^{i\theta_1}, \dots)| \, d\theta_1 \cdots$. The numerical value is approximately $0.54441...\dots$.
**Task:** Find a symbolic closed-form expression for $m(1+x+y+z+w)$ without using double L-values of modular forms, single L-values, or the Chowla-Selberg period.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | number_theory | ground_truth_computable | 2 | 0.54441256175218558519587806274502767666605280202852627449556789488000645997738563329065126658200759562393248342 | https://dms.umontreal.ca/~mlalin/surveyMahlerfinal-revised.pdf | Bertin & Lalin survey on Mahler measure of multivariable polynomials. The Mahler measure m(1+x+y+z+w) extends Smyth's results to 4 variables with connections to L-functions | null | null | null | null | null |
mahler_elliptic_product | Consider the following research problem in mathematics.
**Mahler Measure of $(x+y+1)(x+1)(y+1)-xy$**
**Definition:** This problem concerns the logarithmic Mahler measure $m(P) = \frac{1}{(2\pi)^2} \int_0^{2\pi} \int_0^{2\pi} \log |P(e^{i\theta}, e^{i\phi})| \, d\theta \, d\phi$ of the two-variable Laurent polynomial $P(x,y) = (x+y+1)(x+1)(y+1) - xy$. This polynomial belongs to a genus-one (elliptic) Mahler-measure family studied via regulators and $q$-series methods, but an explicit closed form in standard special functions is not known. The numerical value is approximately $0.66422...\dots$.
**Task:** Find a symbolic closed-form expression for $m((x+y+1)(x+1)(y+1)-xy)$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | number_theory | ground_truth_computable | 2 | 0.66422509302916593526284646964035380327719614159380234519653938087512261465036362537617710889395147153204690603639639539212919594553663512901466775635 | https://arxiv.org/abs/1012.3036 | Rogers and Zudilin: 'From L-series of elliptic curves to Mahler measures'. Studies genus-one Mahler-measure families of product-of-linear-factors type via regulators and q-series methods | null | null | null | null | null |
mzv_reduction_zeta_3_3_3 | Consider the following research problem in mathematics.
**Reduction of $\zeta(3,3,3)$**
**Definition:** The Multiple Zeta Value $\zeta(3,3,3)$ is a depth-3, weight-9 value defined by $\sum_{n_1 > n_2 > n_3 \geq 1} (n_1 n_2 n_3)^{-3}$. The problem is to determine if and how this value can be expressed in terms of lower-depth MZVs or products of standard zeta values.
**Task:** Find a closed-form expression for $\zeta(3,3,3)$ in terms of lower-depth Multiple Zeta Values or standard constants.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | number_theory | ground_truth_computable | 0 | 0.012034182574412003861599684421693740505784954499279660274108607505043368975229731321242723660408603557091175883 | https://arxiv.org/abs/math/0309425 | Hoffman: 'Algebraic Aspects of Multiple Zeta Values'. Establishes algebraic framework for reducing MZVs like zeta(3,3,3) using shuffle/stuffle algebra relations | null | null | null | null | null |
stieltjes_gamma_1 | Consider the following research problem in mathematics.
**Closed Form for Stieltjes Constant $\gamma_1$**
**Definition:** The Stieltjes constants $\gamma_n$ are the coefficients in the Laurent series expansion $\zeta(1+s) = \frac{1}{s} + \sum_{n \geq 0} \frac{(-1)^n}{n!} \gamma_n s^n$ of the Riemann zeta function $\zeta(s)$ about $s=1$. $\gamma_1$ has a value of approximately $-0.07281...$. It is unknown if $\gamma_1$ can be expressed in terms of classical constants or if it represents a new fundamental constant.
**Task:** Find a symbolic closed-form expression for the Stieltjes constant $\gamma_1$.
Even if your solution is unproven, you must propose an accurate candidate solution (such as a conjecture) that satisfies the requirements listed above and adequately addresses the task. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | number_theory | ground_truth_computable | 0 | -0.072815845483676724860586375874901319137736338334337952599006559741401433571511484878086928244844014604077207279 | https://oeis.org/A082633 | OEIS provides an entry for the decimal expansion of the 1st negated Stieltjes constant gamma_1. It also cites Maślanka, K., & Koleżyński, A. (2022). The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm. Computational Methods in Science & Technology, 28(2) to provide 0.072815... | null | null | null | null | null |
closed_form_ramanujan_soldner | Consider the following open problem.
**Closed-Form Expression for the Ramanujan-Soldner Constant (μ)**
**Definition:** μ is the unique positive real number satisfying li(μ)=0, where li is the non-offset logarithmic integral (Cauchy principal value). Equivalently, li(x)=Ei(log x) for x>0.
**Task:** Find a finite explicit expression for μ in terms of other known constants and standard special functions available in mpmath.
Operational constraints:
- The result must be a finite expression tree over allowed primitives.
- No numerical root-finding (findroot), quadrature (quad), nsum/nprod, loops implementing approximations, or hard-coded high-precision decimal literals.
- Do not restate μ implicitly as the solution of an equation.
- Inverses are allowed only if they are standard named functions directly available in mpmath.
REQUIRED OUTPUT FORMAT:
def proposed_solution():
from mpmath import mp
mp.dps = 100
result = ...
return result | constant | number_theory | ground_truth_computable | 2 | 1.45136923488338105028396848589202744949303228 | https://oeis.org/A070769 | OEIS A070769: Ramanujan-Soldner constant μ, the unique positive zero of li(x). See also MathWorld and Wikipedia for definition and properties. | null | null | null | null | null |
schur_6 | Let S(k) be the Schur number: the largest n such that {1,2,...,n} can be partitioned into k sum-free sets. A subset A of positive integers is sum-free if there do not exist x,y in A with x+y in A (x and y may be equal).
Task: Construct a 6-coloring of {1,2,...,N} with no monochromatic solution to x+y=z (equivalently, each color class is sum-free), and make N as large as possible.
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Return a coloring of {1,...,N} into 6 sum-free color classes.
# colors[0] = 0 (unused), colors[i] in {0,1,2,3,4,5} for i = 1..N.
return {"coloring": [0, c_1, c_2, ..., c_N]}
```
**Validity:** For every color c and all x, y with 1 <= x <= y <= N, if colors[x] = colors[y] = c and x+y <= N, then colors[x+y] != c.
**Score:** maximize N. | construction | combinatorics | benchmark_best_known | 2 | null | https://www.combinatorics.org/ojs/index.php/eljc/article/view/v7i1r32 | Fredricksen & Sweet (2000) give explicit constructions proving S(6)≥536. Later work notes only bounds are 536≤S(6)≤1836, so the optimum is unknown. | null | N | maximize | 536 | null |
euler_mascheroni_closed_form | Consider the following research problem in mathematics.\n\n**Closed-Form Expression for the Euler-Mascheroni Constant**\n\n**Definition:** The Euler-Mascheroni constant is \(\gamma = \lim_{n\to\infty}(\sum_{k=1}^n 1/k - \log n)\). Although many representations are known (limits, integrals, series), no closed-form expression in standard constants/functions is known.\n\n**Task:** Propose a candidate closed-form expression for \(\gamma\) as a finite combination of allowed constants and functions. Your proposal may be conjectural, but it must numerically match \(\gamma\) to high precision.\n\n**Restrictions (enforced by policy/compliance checks):**\n- Do NOT use \`mp.euler\`, \`mp.eulerconst\`, \`mp.digamma\`/\`mp.psi\`, or any function that directly encodes \(\gamma\).\n- Do NOT use numerical quadrature, root-finding, infinite sums/products, \`nsum\`, \`nprod\`, loops that implement a truncation scheme, or limits.\n- Do NOT hardcode the decimal expansion of \(\gamma\) (or return a string of digits).\n- You may use only fixed finite compositions of: \`mp.pi\`, \`mp.e\`, rational numbers, \`mp.log\`, \`mp.sqrt\`, \`mp.exp\`, \`mp.gamma\` evaluated at rational arguments, \`mp.zeta\` evaluated at positive integers \(\ge 2\), \`mp.polylog\` at algebraic arguments, and other standard special functions PROVIDED they do not trivially encode \(\gamma\).\n\nREQUIRED OUTPUT FORMAT:\n\n```python\ndef proposed_solution():\n from mpmath import mp\n mp.dps = 100\n result = ...\n return result\n``` | constant | number_theory | ground_truth_computable | 3 | 0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495 | https://www.ams.org/bull/2013-50-04/S0273-0979-2013-01423-X/ | Lagarias (Bull. AMS, 2013) surveys Euler's constant and modern developments; key arithmetic questions and the absence of a known closed-form expression remain open. Decimal expansion is standard; see OEIS A001620. | null | null | null | null | null |
elliptic_curve_rank_30 | Consider the following optimization problem.
**Elliptic Curve with Rank at Least 30**
**Definition:** The rank of an elliptic curve $E$ over $\mathbb{Q}$ measures the number of independent rational points of infinite order. An elliptic curve with rank at least 29 is known; and under GRH the rank is exactly 29, achieved by Elkies and Klagsbrun in August 2024, breaking the previous record of 28 that stood since 2006. It is unknown whether elliptic curve ranks over $\mathbb{Q}$ can be arbitrarily large.
**Task:** Construct an elliptic curve $E: y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$ over $\mathbb{Q}$ with rank at least 30, along with 30 independent rational points of infinite order.
**Current State-of-the-Art:**
- Metric: rank of an elliptic curve over Q
- Best Known Value: 29
- Direction: MAXIMIZE (higher is better)
- Source: Z29 in E(Q) (2024) by Noam D. Elkies and Zev Klagsbrun
**REQUIRED OUTPUT FORMAT:**
Return your solution as a Python function that returns a dictionary:
def proposed_solution():
# Your solution code here
return {
"curve": [a1, a2, a3, a4, a6], # Weierstrass coefficients (integers or ratio strings \"p/q\")
"points": [[x1, y1], [x2, y2], ...] # at least 30 rational points of infinite order; coordinates must be integers or ratio strings \"p/q\"
}
**To beat the baseline:** Your result must be > 29
| construction | number_theory | benchmark_best_known | 2 | null | https://arxiv.org/abs/0709.2908 | Noam Elkies, 'Three lectures on elliptic surfaces and curves of high rank' (2007). Documents the rank 28 record from 2006; note that Elkies-Klagsbrun found rank 29 in 2024 (no single arXiv paper yet, but announced August 2024). | null | null | null | null | null |
elliptic_curve_rank_torsion_z7z | Consider the following optimization problem.
**High-Rank Elliptic Curve with Torsion $\mathbb{Z}/7\mathbb{Z}$**
**Definition:** For elliptic curves over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/7\mathbb{Z}$, the current rank record is 6. Finding curves with higher rank and prescribed torsion is a major challenge, as heuristics suggest the maximum achievable rank decreases with larger torsion.
**Task:** Construct an elliptic curve over $\mathbb{Q}$ with torsion subgroup exactly $\mathbb{Z}/7\mathbb{Z}$ and rank at least 7, providing 7 independent points of infinite order plus a point of order 7.
**Current State-of-the-Art:**
- Metric: rank of elliptic curve
- Best Known Value: 6
- Direction: MAXIMIZE (higher is better)
- Source: New Rank Records For Elliptic Curves Having Rational Torsion (2020) by Noam D. Elkies and Zev Klagsbrun
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Your solution code here
return {
"curve": [a1, a2, a3, a4, a6], # Weierstrass coefficients (integers or ratio strings \"p/q\")
"torsion_point": [x, y], # A point of order 7 on the curve (integers or ratio strings \"p/q\")
"infinite_order_points": [[x1, y1], [x2, y2], ...] # Points of infinite order; coordinates must be integers or ratio strings \"p/q\"
}
**To beat the baseline:** Your result must be > 6
| construction | number_theory | benchmark_best_known | 2 | null | https://arxiv.org/abs/2003.00077 | Elkies and Klagsbrun, 'New Rank Records For Elliptic Curves Having Rational Torsion' (2020). Presents rank-record breaking elliptic curves with torsion subgroups including Z/7Z (current record rank >= 6 by Klagsbrun). | null | null | null | null | null |
sum_three_cubes_114 | **Sum of Three Cubes for $n = 114$**
**Definition:** The equation $x^3 + y^3 + z^3 = n$ asks whether an integer $n$ can be expressed as a sum of three integer cubes. After the solutions for 33 and 42 were found in 2019, only seven integers below 1000 remain unsolved: 114, 390, 627, 633, 732, 921, and 975. No solutions are known despite extensive computational searches.
**Task:** Find integers $x, y, z$ such that $x^3 + y^3 + z^3 = 114$.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Your solution code here
return {
"x": <int>, # Integer x
"y": <int>, # Integer y
"z": <int> # Integer z
}
# Alternative format: return [x, y, z] | construction | number_theory | new_construction | 1 | null | https://oeis.org/A060464 | OEIS A060464: Integers that potentially can be represented as sums of three cubes. After solving 33 and 42, 114 is the smallest remaining unsolved case as of 2025. References Booker-Sutherland computations. | null | null | null | null | null |
sum_three_cubes_390 | **Sum of Three Cubes for $n = 390$**
**Definition:** The equation $x^3 + y^3 + z^3 = n$ asks whether an integer $n$ can be expressed as a sum of three integer cubes. The integer 390 is one of seven remaining unsolved cases below 1000. Since $390 \equiv 3 \pmod 9$, a solution is not ruled out by congruence conditions.
**Task:** Find integers $x, y, z$ such that $x^3 + y^3 + z^3 = 390$.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
return {
"x": <int>, # Integer x
"y": <int>, # Integer y
"z": <int> # Integer z
}
# Alternative format: return [x, y, z] | construction | number_theory | new_construction | 1 | null | https://arxiv.org/pdf/2007.01209 | Booker and Sutherland (2020). 'On a question of Mordell.' Lists 390 among unresolved values ≤1000 at that time, and describes very large searches for solutions (including ruling out solutions with small “min(|x|,|y|,|z|)” up to huge bounds) | null | null | null | null | null |
sum_three_cubes_627 | **Sum of Three Cubes for $n = 627$**
**Definition:** The integer 627 is one of seven remaining integers below 1000 for which no representation as a sum of three cubes is known. Since $627 \equiv 6 \pmod 9$, congruence conditions do not rule out a solution.
**Task:** Find integers $x, y, z$ such that $x^3 + y^3 + z^3 = 627$.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
return {
"x": <int>, # Integer x
"y": <int>, # Integer y
"z": <int> # Integer z
}
# Alternative format: return [x, y, z] | construction | number_theory | new_construction | 1 | null | https://arxiv.org/abs/1903.04284 | Booker (2019). 'Cracking the problem with 33.' Lists 390 among the seven remaining unsolved cases under 1000 (114, 390, 627, 633, 732, 921, 975). No representation as sum of three cubes is known. | null | null | null | null | null |
sum_three_cubes_primitive_192 | **Primitive Sum of Three Cubes for $n = 192$**
**Definition:** While $192=4^3+4^3+4^3$ admits a non-primitive solution with $\text{gcd}(x,y,z)=4,$, no primitive solution (where $\gcd(x,y,z) = 1$) is known for $x^3 + y^3 + z^3 = 192$.
**Task:** Find integers $x, y, z$ with $\gcd(x, y, z) = 1$ such that $x^3 + y^3 + z^3 = 192$.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
return {
"x": <int>, # Integer x
"y": <int>, # Integer y
"z": <int> # Integer z (must have gcd(x,y,z) = 1)
}
# Alternative format: return [x, y, z] | construction | number_theory | new_construction | 1 | null | https://oeis.org/A060464 | OEIS sequence on sums of three cubes; references Elsenhans & Jahnel (2009) showing 192, 375, 600 have no known primitive solutions with gcd(x,y,z)=1 | null | null | null | null | null |
mahler_x_3_y_3_1_5xy | Consider the following research problem in mathematics.
**Mahler Measure of $x^3+y^3+1-5xy$**
**Definition:** This problem concerns the logarithmic Mahler measure of the polynomial $Q_5(x, y) = x^3 + y^3 + 1 - 5xy$. This polynomial belongs to the Hesse family $Q_k(x, y) = x^3 + y^3 + 1 - kxy$, whose Mahler measures are related in special cases to $L$-values of modular forms (e.g., $k=3$ yields a dilogarithm value), but no general closed form is known for generic $k$. The numerical value is approximately $1.5923...\dots$.
**Task:** Find a symbolic closed-form expression for the logarithmic Mahler measure $m(Q_5) = m(x^3+y^3+1-5xy)$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | number_theory | ground_truth_computable | 0 | 1.5923685610864577552648762016584343966931986506568980628466025871066531426921883851477685159655913223305979340 | https://arxiv.org/abs/math/0308041 | Rogers (2010), 'Hypergeometric formulas for lattice sums and Mahler measures.' Provides a general hypergeometric formula for $Q_k(x, y)=x^3+y^3+1-kxy. | null | null | null | null | null |
c5_ising_susceptibility | Consider the following research problem in mathematics.
**Closed Form for the 5th Ising Susceptibility Integral ($C_5$)**
**Definition:** The integrals $C_n$ appear in the susceptibility expansion of the 2D Ising model and are defined as: $C_n = \frac{2^n}{n!} \int_0^\infty t K_0(t)^n dt$ where $K_0(t)$ is the modified Bessel function of the second kind. While $C_1$ through $C_4$ have known closed forms involving $\pi$, Catalan's constant $G$, and other standard constants, $C_5$ (approx.\ $0.66575...\dots$) remains unsolved. It is conjectured to be a period of a Calabi-Yau 3-fold.
**Task:** Find a symbolic closed-form expression for the Ising integral $C_5$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.66575980019993742831573380830706659819749638207949765953944270353122704376721234786771901508036929308584399492431185604034925933005075368056386687474090556074714047548823410663129381029978766539289878 | https://www.davidhbailey.com/dhbpapers/ising.pdf | Bailey, Borwein, Crandall, 'Integrals of the Ising class' (2006) - provides a definition for these Ising integrals and high-precision numerical results | null | null | null | null | null |
c6_ising_susceptibility | Consider the following research problem in mathematics.
**Closed Form for the 6th Ising Susceptibility Integral ($C_6$)**
**Definition:** The integrals $C_n$ appear in the susceptibility expansion of the 2D Ising model and are defined as: $C_n = \frac{2^n}{n!} \int_0^\infty t K_0(t)^n dt$ where $K_0(t)$ is the modified Bessel function of the second kind. $C_6$ has a numerical value of approximately $0.64863\dots$. No closed-form expression is currently known.
**Task:** Find a symbolic closed-form expression for the Ising integral $C_6$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.64863420903100707526314984345035169088977250948162799561505088718478178178800557923682516243508678874630577856026398027701536062285107772881321904645186423022491587784838301747 | https://www.davidhbailey.com/dhbpapers/ising.pdf | Bailey, Borwein, Crandall, 'Integrals of the Ising class' (2006) - provides a definition for these Ising integrals and high-precision numerical results | null | null | null | null | null |
c7_ising_susceptibility | Consider the following research problem in mathematics.
**Closed Form for the 7th Ising Susceptibility Integral ($C_7$)**
**Definition:** The integrals $C_n$ appear in the susceptibility expansion of the 2D Ising model and are defined as: $C_n = \frac{2^n}{n!} \int_0^\infty t K_0(t)^n dt$ where $K_0(t)$ is the modified Bessel function of the second kind. $C_7$ has a numerical value of approximately $0.63997\dots$.
**Task:** Find a symbolic closed-form expression for the Ising integral $C_7$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.63997304682795750054991340799259099278899717666159325886302862532801001076106427 | https://www.davidhbailey.com/dhbpapers/ising.pdf | Bailey, Borwein, Crandall, 'Integrals of the Ising class' (2006) - provides a definition for these Ising integrals and high-precision numerical results | null | null | null | null | null |
calabi_yau_c5 | Consider the following research problem in mathematics.
**Structural Identification of the Calabi-Yau Variety for $C_5$**
**Definition:** The Ising susceptibility integral $C_5$ is conjectured to be a period of a specific Calabi-Yau 3-fold. This structural connection suggests that $C_5$ can be represented via the geometry of a specific algebraic variety, but the explicit defining equations of this variety are unknown.
**Task:** Identify the explicit defining equations of the Calabi-Yau variety whose period equals $C_5$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | 9586.9411228790989677465668396217590140439479019447662973679749308496694302478578092951538171573178204361535269 | https://arxiv.org/abs/1007.0535 | Bostan et al., 'The Ising model: from elliptic curves to modular forms and Calabi-Yau equations' (2010) - Calabi-Yau differential equations emerging in Ising susceptibility analysis | null | null | null | null | null |
mzv_decomposition_c5 | Consider the following research problem in mathematics.
**Multiple Zeta Value Decomposition of $C_5$**
**Definition:** The Ising susceptibility integrals are believed to belong to the algebra of Multiple Zeta Values (MZVs). While the structure is known for small $n$, the specific weight and depth decomposition for $C_5$ within the MZV algebra is an open problem.
**Task:** Determine the specific Multiple Zeta Value decomposition (linear combination of MZVs) that evaluates to $C_5$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | number_theory | ground_truth_computable | 2 | 0.6657598001999374283157338083070665981974963820794976595394427035312270437672123478677190150803692930858440 | https://arxiv.org/abs/0907.2557 | Blumlein, Broadhurst, Vermaseren, 'The Multiple Zeta Value Data Mine' (2009) - proven MZV reductions relevant to physics integrals including Ising-class | null | null | null | null | null |
feynman_3loop_sunrise | Consider the following research problem in mathematics.
**3-Loop Sunrise Diagram at Threshold**
**Definition:** This problem concerns the 3-loop sunrise (banana) Feynman diagram with 4 equal-mass propagators evaluated at threshold $s = 16m^2$. In the position-space Bessel representation, the integral is $B(4) = \int_0^\infty r\, I_0(4r)\, K_0(r)^4\, dr$, where $I_0$ and $K_0$ are modified Bessel functions of order 0. The integral is a period of a Calabi-Yau 2-fold (K3 surface).
**Task:** Find a symbolic closed-form expression for the 3-loop sunrise integral at the threshold $s = 16m^2$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | 2.27729529146683223972828877133800817650258821452965244985120378395321356945250809311211331151764131842932 | https://link.springer.com/content/pdf/10.1007/JHEP05%282021%29066.pdf | Bönisch, Fischbach, Klemm, Nega, Safari (2021). 'Analytic structure of all loop banana integrals' - Eq. (2.10) gives the D=2 Bessel representation. | null | null | null | null | null |
feynman_4loop_banana | Consider the following research problem in mathematics.
**4-Loop Banana Diagram at Threshold**
**Definition:** This problem concerns the 4-loop banana graph with equal masses at the corresponding threshold, $$B(5) = \int_0^{\infty} r \, I_0(5r) \, K_0(r)^5 \, dr,$$ where $I_0$ and $K_0$ are modified Bessel functions of order 0. As the loop order increases, the associated geometry becomes more complex (higher-dimensional Calabi-Yau varieties).
**Task:** Find a symbolic closed-form expression for the 4-loop banana integral at threshold.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | 3.5649669441225491856098202100926563331364799751675362407992703859275965557517521603709835573861024583018782717 | https://link.springer.com/content/pdf/10.1007/JHEP05%282021%29066.pdf | Bönisch, Fischbach, Klemm, Nega, Safari (2021). 'Analytic structure of all loop banana integrals' - Eq. (2.10) gives the D=2 Bessel representation. Eq. (2.10) with their notation gives a prefactor of 16, while our numeric value matches the integral without the prefactor 16 evaluated at threshold. | null | null | null | null | null |
elliptic_kernel_f2_001 | Consider the following open problem in mathematical physics.
**Elliptic-Kernel Log-Moment Constant f2(0,0,1)**
We define the complete elliptic integral of the first kind K(m) for complex parameter m by
K(m) = ∫_{0}^{π/2} dθ / sqrt(1 - m sin^2 θ),
using the principal branch of the square root and analytic continuation in m.
For real s in (1,9), define r = sqrt(s) and A(s) = (r+3)(r-1)^3. For integer m ≥ 1 define
D_m(s) = 2 / sqrt(A(s)) * K( m - 1 - (2m-3)*((r-3)(r+1)^3)/((r+3)(r-1)^3) ).
Let D_1(s) denote D_m(s) at m=1.
For integers i,j,k ≥ 0 and integer m ≥ 1 define
f_m(i,j,k) = ∫_{1}^{9} ds \cdot D_1(s) \cdot Re( 3^{(m-1)/2} * D_m(s) ) \cdot (s - 9/5) \cdot ln^i(9-s) \cdot ln^j(s-1) \cdot ln^k(s),
where Re(\cdot) denotes the real part and ln denotes the principal real logarithm on positive arguments.
**Task:** Find a symbolic closed-form expression for the constant f_2(0,0,1).
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions.
REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100
# Use only mpmath constants/functions and special functions.
# No numerical quadrature, root-finding, or summation loops.
result = ...
return result | constant | continuum_physics | ground_truth_computable | 2 | 30.7476526736391709896774235351358778861783865155459326024781812950213971132375910461620684439641407962420702403407811170933205901539809821596 | https://pos.sissa.it/290/077/pdf | Several other sources reference this quantity: https://pos.sissa.it/303/073/pdf, https://arxiv.org/pdf/1704.06996, and https://arxiv.org/pdf/1910.01248. See equations 23 to 24 in the source_url paper. | null | null | null | null | null |
tracy_widom_f2_mean | Consider the following research problem in mathematics.
**Mean of the Tracy-Widom $F_2$ Distribution**
**Definition:** The Tracy-Widom distribution $F_2$ is the cumulative distribution function (CDF) of a real-valued random variable $X$ describing the fluctuations of the largest eigenvalue of GUE random matrices (after standard centering/scaling). It can be characterized via the Hastings--McLeod solution $q(s)$ of the Painlev\'e II equation
\[ q\''(s) = s\,q(s) + 2q(s)^3, \qquad q(s) \sim \mathrm{Ai}(s) \text{ as } s \to +\infty. \]. Define the mean of the Tracy--Widom $F_2$ law as
\[ \mu_2 := \mathbb{E}[X]. \].
**Task:** Find a symbolic closed-form expression for $\mu_2 = \mathbb{E}[X]$.
This constant appears fundamental in random matrix theory but has resisted identification with known constants despite extensive numerical searches.
Your solution must be a finite combination of:
- Rational and algebraic numbers
- Constants: $\pi$, $e$, $\gamma$, $G$, $\zeta(n)$
- Special functions: $\Gamma$, polylogarithms, elliptic integrals at algebraic arguments
- Painlevé connection constants (if explicitly computable)
INADMISSIBLE:
- The Painlevé II solution itself
- Unevaluated integrals involving $q(s)$
- Numerical approximations
REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | -1.77108680741160162612693822832370833445514095085934616781672203 | https://arxiv.org/abs/0804.2543 | Folkmar Bornemann, 'On the Numerical Evaluation of Fredholm Determinants' (2010). Math. Comp. 79(270):871-915. Provides accurate algorithms for numerical evaluation of Tracy-Widom distributions including mean (approx -1.7711) and variance for F2 (GUE). | null | null | null | null | null |
tracy_widom_f2_variance | Consider the following research problem in mathematics.
**Variance of the Tracy-Widom $F_2$ Distribution**
**Definition:** The variance of the Tracy-Widom $F_2$ distribution is: \[ \mathrm{Var}[X] = \mathbb{E}[X^2] - \mathbb{E}[X]^2 = 0.81319... \] where $X \sim F_2$ with the random-matrix limit definition and standard Airy-kernel normalization. Like the mean, no closed-form expression is known.
**Task:** Find a symbolic closed-form expression for the variance of the Tracy-Widom $F_2$ distribution.
Your solution must be a finite combination of standard constants and special functions, not the defining Painlevé integral.
REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | continuum_physics | ground_truth_computable | 2 | 0.8131947928329 | https://arxiv.org/abs/0904.1581 | Folkmar Bornemann, 'On the Numerical Evaluation of Distributions in Random Matrix Theory' (2010). Provides algorithms to compute variance (approx 0.8132) and other moments of Tracy-Widom F2 distribution. | null | null | null | null | null |
tracy_widom_f1_mean | Consider the following research problem in mathematics.
**Mean of the Tracy-Widom $F_1$ Distribution (GOE)**
**Definition:** Let $q(s)$ be the Hastings--McLeod solution of Painlev\'e II, $q\''(s)=s q(s)+2 q(s)^3$ with $q(s)\sim\mathrm{Ai}(s)$ as $s\to+\infty$. Define
\[ F_2(s)=\exp\!\left(-\int_s^{\infty}(x-s)q(x)^2\,dx\right), \qquad F_1(s)=\exp\!\left(-\tfrac12\int_s^{\infty}q(x)\,dx\right)\,F_2(s)^{1/2}. \]
Let $X$ be a random variable with CDF $F_1$. Define the mean $\mu_1:=\mathbb{E}[X]$.
Numerically, $\mu_1\approx -1.2065...\ldots$.
**Task:** Find a symbolic closed-form expression for $\mu_1$.
Your solution must be a finite combination of:
- Rational and algebraic numbers
- Constants: $\pi$, $e$, $\gamma$, $G$, $\zeta(n)$
- Special functions: $\Gamma$, polylogarithms, elliptic integrals at algebraic arguments
- Painlev\'e connection constants (only if they are explicitly computable in the above admissible class)
INADMISSIBLE:
- The Painlev\'e II function $q(s)$ itself or any evaluation of $q(s)$
- Unevaluated integrals involving $q(s)$
- Numerical approximations (including hard-coded decimal expansions of $\mu_1$)
- Numerical integration, root finding, summation/product loops, or ODE solvers
REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100
# Use only mpmath constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions, and special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...).
# No numerical integration, root finding, summation/product loops, or ODE solvers.
result = ...
return result | constant | continuum_physics | ground_truth_computable | 2 | -1.206533574582093757882324561830899612811508928919795846796986046439531871428069093892948158498295831217412832146379216871 | https://arxiv.org/abs/0904.1581 | Bornemann, 'On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review' (2009), Example 8.4.1 tabulates the mean of $F_1$ as approximately -1.2065335745820; higher-precision digits here are computed offline using Painlev\'e/Fredholm-determinant methods following Bornemann. | null | null | null | null | null |
monomer_dimer_entropy | Let \Lambda_{m,n} be the m\times n rectangular subgraph of the 2D square lattice with free boundary. A configuration is a matching: a set of disjoint dimers (edges), with all uncovered vertices treated as monomers. Assign weight z to each monomer and weight 1 to each dimer. Define the finite-volume partition function
Z_{m,n}(z) = \sum_{\text{matchings }M} z^{\#\text{monomers}(M)}.
Define the entropy (free energy) per site by the thermodynamic limit
s(z) = \lim_{m,n\to\infty} \frac{1}{mn}\log Z_{m,n}(z),
which is known to exist and to be independent of boundary conditions for this model.
At z=0 (perfect matchings only; take m,n even), s(0) = G/\pi is known (Kasteleyn / Temperley-Fisher).
At z=1 (all matchings equally weighted), the square-lattice monomer-dimer constant is
s(1) = 0.66279...,
and no closed-form expression is known for s(1).
**Task:** Find a symbolic closed-form expression for the monomer-dimer entropy $s(1)$ on the square lattice.
Your solution must be a finite combination of:
- Rational and algebraic numbers
- Constants: $\pi$, $e$, $\gamma$, $G$, $\zeta(n)$
- Special functions at algebraic arguments
INADMISSIBLE:
- The partition function series/limit itself
- Transfer matrix eigenvalues
- Numerical approximations
REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.662798972834 | https://arxiv.org/abs/cond-mat/0610690 | Kong (2006) estimates the square-lattice monomer-dimer constant as h2 = 0.662798972834 (claimed 11 correct digits) and brackets it near 0.662798972831 < h2 < 0.662798972845. Butera et al. (2012, arXiv:1206.0872) summarize tight bounds 0.66279897190 ≤ h2 ≤ 0.662798972844913 and a best estimate h2 = 0.6627989727(1). | null | null | null | null | null |
hard_square_entropy | Consider the following research problem in mathematics.
**Hard Square Entropy Constant**
**Definition:** The hard square model (also called the hard-core lattice gas on $\mathbb{Z}^2$) counts independent sets on the square lattice. Let $F(m,n)$ be the number of $m \times n$ binary matrices with no two adjacent 1s (horizontally or vertically). The hard square entropy constant is defined as:
\[ \kappa = \lim_{n \to \infty} [F(n,n)]^{1/n^2} \approx 1.5030... \]
**Current Status:** Unlike the hard hexagon model on the triangular lattice (which Baxter solved exactly in 1980 using corner transfer matrices and Rogers-Ramanujan identities), the hard square model has NO KNOWN CLOSED FORM despite decades of research.
**Task:** Find a closed-form expression for $\kappa$ in terms of standard mathematical constants and special functions.
**Known bounds:**
- Calkin-Wilf (1998): $1.5030... \leq \kappa \leq 1.5035...$
- Numerical: $\kappa \approx 1.5030...$
- Entropy per site: $\log(\kappa) \approx 0.40749$
**Context:** The hard square model is central to combinatorics (counting independent sets), statistical mechanics (lattice gases), and probability (hardcore processes). A closed form would be a major breakthrough.
Your solution must be a finite symbolic expression involving standard constants (e.g., $\pi$, $e$, $\gamma$, algebraic numbers) and/or special functions (e.g., $\Gamma$, hypergeometric, elliptic integrals, polylogarithms).
INADMISSIBLE:
- Numerical approximations or truncated series
- Transfer matrix eigenvalue formulas
- Infinite products without closed evaluation
REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 1.5030480824753322643220663294755536893857810 | https://oeis.org/A085850 | OEIS A085850: Decimal expansion of hard square entropy constant kappa = 1.503048082475... References Baxter's 'Planar Lattice Gases with Nearest-Neighbour Exclusion' and Finch's 'Mathematical Constants' (2003). | null | null | null | null | null |
saw_square_lattice | Consider the following research problem in mathematics.
**Connective Constant for Square Lattice Self-Avoiding Walks**
**Definition:** A self-avoiding walk (SAW) on a lattice is a path that visits each lattice site at most once. The number of $n$-step SAWs starting from the origin on the square lattice $\mathbb{Z}^2$ is conjectured to grow asymptotically as $c_n \sim A \mu^n n^{11/32}$, where $\mu$ is the connective constant. The value $\mu = \lim_{n \to \infty} c_n^{1/n}$ has been computed to high precision via exact enumeration (to $n = 79$ steps) and series analysis: $\mu = 2.6381...(3)$. Unlike the honeycomb lattice, where Duminil-Copin and Smirnov (2012) proved $\mu = \sqrt{2 + \sqrt{2}}$, no closed-form expression is known for the square lattice.
**Task:** Find a symbolic closed-form expression for the square lattice self-avoiding walk connective constant $\mu = 2.6381...$
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 3 | 2.63815853032790 | https://arxiv.org/pdf/1607.02984 | Jacobsen, Scullard, Guttmann. (2016). Provides a high-precision estimate for the growth constant for square-lattice self-avoiding walks. The best conjecture from Jacobsen-Scullard-Guttmann provide $t = \sqrt{\frac{7 + \sqrt{30261}}{26}} = 2.6381585303417408684\dots$ as their estimate, but it only matches 11 significant digits. We require 13 matching digits to prevent this known-insufficient conjecture from being accepted. | null | null | null | null | null |
saw_triangular_lattice | Consider the following research problem in mathematics.
**Connective Constant for Triangular Lattice Self-Avoiding Walks**
**Definition:** The connective constant $\mu = \lim_{n \to \infty} c_n^{1/n}$ for self-avoiding walks on the triangular lattice governs the exponential growth rate of $n$-step walks: $c_n \sim A \mu^n n^{\gamma-1}$. High-precision series analysis gives $\mu = 4.1507...(26)$. It was conjectured that $\mu_{\text{triangular}} + \mu_{\text{honeycomb}} = 6$, but this has been ruled out numerically. No closed-form expression is known.
**Task:** Find a symbolic closed-form expression for the triangular lattice self-avoiding walk connective constant $\mu = 4.1507...$
Propose your best conjectured symbolic solution, which will be evaluated against a high-precision numerical ground-truth solution. Even if your solution is unproven, you must propose a candidate solution that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 3 | 4.15079722 | https://arxiv.org/abs/cond-mat/0409039 | Iwan Jensen, “Self-avoiding walks and polygons on the triangular lattice,” J. Stat. Mech. (2004) P10008. Reports the estimate as $\mu = 4.150797226(26)$. | null | null | null | null | null |
saw_simple_cubic | Consider the following research problem in mathematics.
**Connective Constant for Simple Cubic Lattice Self-Avoiding Walks**
**Definition:** The connective constant $\mu=\lim_{n \to \inf} c_n^{1/n}$ for self-avoiding walks on the three-dimensional simple cubic lattice $\mathbb{Z}^3$ has been computed via the pivot algorithm to unprecedented precision: $\mu = 4.6840...(27)$ (Clisby, 2013). The longest exactly enumerated walks have 36 steps. No closed-form expression is known for any 3D lattice, and the problem is significantly harder than the 2D case due to the absence of conformal field theory techniques.
**Task:** Find a symbolic closed-form expression for the simple cubic lattice self-avoiding walk connective constant $\mu = 4.6840...$
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 3 | 4.684039931 | https://arxiv.org/abs/1302.2106 | Clisby (2013) 'Calculation of the connective constant for self-avoiding walks on the simple cubic lattice'; mu = 4.684039931(27) | null | null | null | null | null |
madelung_nacl | Consider the following research problem in mathematics.
**Closed Form for the NaCl Madelung Constant**
**Definition:** The Madelung constant $M$ for a crystal structure quantifies the electrostatic energy of an ion in the lattice. For the rock salt (NaCl) structure with alternating positive and negative ions on a cubic lattice, the constant is defined by the conditionally convergent sum: \[ M = \sum_{(i,j,k) \neq (0,0,0)} \frac{(-1)^{i+j+k}}{\sqrt{i^2 + j^2 + k^2}} \] where $M$ is the magnitude of the Madelung constant normalized by nearest-neighbor distance and obtained by analytic continuation / the accepted bulk limit (average of Evjen even/odd limits). The value is $M = 1.7475...$. Despite over a century of study since Madelung (1918), (Bailey et al., 2006).
**Task:** Find a symbolic closed-form expression for the NaCl Madelung constant $M = 1.7475...$
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 1.7475645946331821906362120355443974034851614366247417581528 | https://oeis.org/A085469 | OEIS decimal expansion of negated Madelung constant for NaCl structure; value approximately 1.7475645946...; no closed form known (Bailey et al. 2006) | null | null | null | null | null |
madelung_cscl | Consider the following research problem in mathematics.
**Closed Form for the CsCl Madelung Constant**
**Definition:** The Madelung constant for the cesium chloride (CsCl) structure, where each ion is surrounded by 8 nearest neighbors of opposite charge in a body-centered cubic arrangement, is $M = 1.7626...$. The lattice sum is: $M = \frac{2}{\sqrt{3}} \sideset{}{^{\mathrm{Ewald}}}{\sum}_{(i,j,k) \in \mathbb{Z}^3} \frac{(-1)^{i+j+k}}{\sqrt{(i - \tfrac{1}{2})^2 + (j - \tfrac{1}{2})^2 + (k - \tfrac{1}{2})^2}}$ where the Ewald sum denotes the Ewald-summed (order-independent) value of this conditionally convergent Coulomb lattice sum, and the prefactor $2/\sqrt{3}$ fixes the normalization to the nearest-neighbour distance convention. The sum converges even more slowly than NaCl due to the BCC geometry.
**Task:** Find a symbolic closed-form expression for the CsCl Madelung constant $M = 1.7626...$
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 1.76267477307098839793567332063864429117052861958858528064941843772796622376934083047150945811216988908569 | https://oeis.org/A181152 | OEIS decimal expansion of the (magnitude of the) CsCl Madelung constant; OEIS describes it as 'negated' under a common sign convention, but this benchmark uses the positive magnitude $M \approx 1.7627$. | null | null | null | null | null |
madelung_zns | Consider the following research problem in mathematics.
**Closed Form for the Zincblende (ZnS) Madelung Constant**
**Definition:** The Madelung constant for the zincblende (sphalerite) structure, adopted by ZnS and many III-V semiconductors, is $M = 1.6380...$. In this structure, each ion has 4 nearest neighbors in a tetrahedral arrangement. The lattice sum is conditionally convergent and computed via Ewald summation. .
**Task:** Find a symbolic closed-form expression for the zincblende Madelung constant $M = 1.6380...$
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 1.638055053388789423750034776358619465360179663136657883957644623927706812837223137698546420043494665161 | https://oeis.org/A182566 | OEIS decimal expansion of negated Madelung constant for zincblende (sphalerite) ZnS; value 1.6380550533887894... | null | null | null | null | null |
site_percolation_square | Consider the following research problem in mathematics.
**Site Percolation Threshold on the Square Lattice**
**Definition:** Consider independent nearest-neighbor site percolation on $\mathbb{Z}^2$ (the infinite square lattice): each vertex is independently declared 'open' with probability $p$ and 'closed' with probability $1-p$. Two open sites are connected if they are nearest neighbors (adjacent vertices, i.e., distance 1 apart in the lattice). Define $p_c = \inf\{p \in [0,1] : P_p(\exists \text{ an infinite connected cluster of open sites}) > 0\}$. Note: this is site percolation, not bond percolation. Bond percolation on $\mathbb{Z}^2$ has exact $p_c = 1/2$ by Kesten (1980), but the self-duality argument does not apply to site percolation.
**Task:** Find a symbolic closed-form expression for the site percolation threshold $p_c$ on $\mathbb{Z}^2$.
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | lattice_models | ground_truth_computable | 2 | 0.59274605079210 | https://iopscience.iop.org/article/10.1088/1751-8113/48/45/454003/pdf | Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley-Lieb algebras'. Approximately 14 reliable digits. No closed form or conjecture known. | null | null | null | null | null |
knot_volume_6_3 | Consider the following research problem in mathematics.
**Hyperbolic Volume of the $6_3$ Knot**
**Definition:** The complement of the knot $6_3$ in the 3-sphere is a hyperbolic 3-manifold with a finite volume (approximately $5.7760...\dots$). The volume is known to be expressible as a sum of Bloch\u2013Wigner dilogarithms at algebraic arguments derived from the ideal triangulation gluing equations (computed routinely by SnapPy/SnapPea). What remains open is whether this volume admits a simpler closed-form expression in terms of standard mathematical constants such as $\pi$, $\zeta(n)$, or $L$-values of number fields.
**Task:** Find a symbolic closed-form expression for the hyperbolic volume of the $6_3$ knot complement in terms of standard mathematical constants (beyond the known Bloch\u2013Wigner dilogarithm representation).
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.polylog, mp.ellipk, mp.hyper, ...)
# No other functions are permitted.
result = ... # your closed-form expression
return result | constant | discrete_geometry | ground_truth_computable | 3 | 5.693021091281300765112483277481222926944301733006880037850870699995476072590906707654919542407040036141224456802400770331855359928066927002673172155677 | https://katlas.org/wiki/6_3 | R.M. Kashaev's 1996 paper 'The hyperbolic volume of knots from quantum dilogarithm' (arXiv:q-alg/9601025, Lett. Math. Phys.). Establishes the fundamental connection between hyperbolic volumes of knot complements and quantum dilogarithm expressions. Provides the mathematical framework for understanding why finding explicit dilogarithm arguments for knot volumes (like the 6_3 knot) is an important research problem. The Knot Atlas website provides a numerical value for the quantity. | null | null | null | null | null |
lattice_packing_dim10 | Consider the following optimization problem.
**Improve a 10D Lattice Packing (Λ10 Baseline)**
**Definition:** A lattice in $\mathbb{R}^{10}$ is $L = \{ z^T B : z \in \mathbb{Z}^{10}\}$ where $B$ is a $10\times 10$ basis matrix (rows are basis vectors). Let $\lambda_1(L)$ be the shortest nonzero vector length and $\operatorname{covol}(L)=|\det(B)|$.
Packing radius: $r=\lambda_1(L)/2$.
Packing density:
\[\delta=\frac{\operatorname{Vol}(B_{10}(r))}{\operatorname{covol}(L)}.\]
**Task:** Construct a lattice in $\mathbb{R}^{10}$ with packing density strictly greater than the best known value.
**Current State-of-the-Art:**
- Metric: `packing_density`
- Best Known Value: `0.092021...`
- Direction: MAXIMIZE
- Best known construction: the laminated lattice $\Lambda_{10}$ (LAMBDA10), with covolume $16\sqrt{3}$ and shortest vector length 2.
- Proven optimal? **No** (optimality in dimension 10 is open).
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Return a 10x10 basis matrix whose rows are basis vectors in R^10
return {
"basis": [
[b_11, b_12, ..., b_1_10],
[b_21, b_22, ..., b_2_10],
# ... 10 rows total
[b_10_1, b_10_2, ..., b_10_10]
]
}
```
**To beat the baseline:** validator-computed `packing_density` must be `> 0.092021...`. | construction | discrete_geometry | benchmark_best_known | 2 | null | https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA10.html | One can also compute this by noting that the laminated lattice Λ10 has Gram matrix determinant 768, so covolume = sqrt(768) = 16√3, shortest vector length 2, packing radius 1, and density Vol(B_10(1))/(16√3) = π^5/(1920√3) ≈ 0.09202111843130556. This is from RWTH Aachen “Catalogue of Lattices” entry for LAMBDA10. Brouwer's note on 'Lattices' in 2002 also provides general background: https://aeb.win.tue.nl/latt/lattices.pdf. | null | null | null | null | null |
periodic_packing_dim10 | Consider the following optimization problem.
**Improve a 10D Periodic Packing (P10c Baseline)**
**Definition:** A periodic packing is a finite union of lattice translates:
\[P = \bigcup_{i=1}^k (L + s_i),\]
where $L\subset\mathbb{R}^{10}$ is a lattice and $s_1,\dots,s_k\in\mathbb{R}^{10}$ are shift vectors (with $s_1=0$ by convention). Let $d_{\min}$ be the minimum distance between distinct packing centers (including across lattice translates). The packing radius is $r=d_{\min}/2$ and density is:
\[\delta(P)=\frac{k\,\operatorname{Vol}(B_{10}(r))}{\operatorname{covol}(L)}.\]
**Task:** Construct a periodic packing in $\mathbb{R}^{10}$ with density strictly greater than the best known value.
**Current State-of-the-Art:**
- Metric: `packing_density`
- Best Known Value: `0.099615...`
- Direction: MAXIMIZE
- Best known construction: Best's $P_{10c}$, obtained via Construction A from a $(10,40,4)$ binary code ($k=40$ cosets of $2\mathbb{Z}^{10}$).
- Proven optimal? **No** (optimality in dimension 10 is open).
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Return a 10x10 lattice basis (rows are basis vectors) and k shift vectors.
# Convention: shifts[0] must be the all-zero vector.
return {
"basis": [
[b_11, b_12, ..., b_1_10],
# ... 10 rows
[b_10_1, b_10_2, ..., b_10_10]
],
"shifts": [
[0.0, 0.0, ..., 0.0],
[s_2_1, s_2_2, ..., s_2_10],
# ... k rows total
[s_k_1, s_k_2, ..., s_k_10]
]
}
```
**To beat the baseline:** validator-computed `packing_density` must be `> 0.099615...`. | construction | discrete_geometry | benchmark_best_known | 2 | null | https://ir.cwi.nl/pub/6831/6831D.pdf | Best constructs a (10,40,4) binary code; applying Construction A yields a 10D periodic packing with center density 40/1024 = 5/128 and packing density (5/128)*Vol_10(1) ≈ 0.0996157828077088. | null | null | null | null | null |
lattice_packing_dim12 | Consider the following optimization problem.
**Dense Lattice Packing in Dimension 12 ($LPD-12$)**
**Definition:** The sphere packing problem in $\mathbb{R}^{12}$. The current best known lattice is $K_{12}$ with packing density 0.0494.
**Task:** Construct a lattice in $\mathbb{R}^{12}$ with a packing density strictly greater than 0.0494.
**Current State-of-the-Art:**
- Metric: sphere packing density
- Best Known Value: 0.049454...
- Direction: MAXIMIZE (higher is better)
- Source: The Coxeter-Todd lattice, the Mitchell group, and related sphere packings (1983) by J. H. Conway and N. J. A. Sloane
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
# Return a 12x12 basis matrix for the lattice, where 'basis' means the 12 row vectors generating the lattice
return {
"basis": [
[b_11, b_12, ..., b_1_12],
[b_21, b_22, ..., b_2_12],
# ... 12 rows total, each with 12 entries
[b_12_1, b_12_2, ..., b_12_12]
]
}
```
**To beat the baseline:** Your result must be > 0.049454...
| construction | discrete_geometry | benchmark_best_known | 2 | null | https://arxiv.org/abs/math/0503446 | Nebe (2005) 'Low dimensional strongly perfect lattices I: The 12-dimensional case' - proves Coxeter-Todd lattice K12 is unique strongly perfect lattice in dimension 12 with densest known packing | null | null | null | null | null |
kissing_number_dim5 | Consider the following optimization problem.
**Kissing Number in Dimension 5**
**Definition:** The kissing number is the maximum number of unit spheres that can touch a central unit sphere. In 5 dimensions, the known bounds are $40 \le \tau_5 \le 44$. The exact value is unknown.
**Task:** Construct a valid kissing configuration in $\mathbb{R}^5$ with strictly more than 40 spheres.
**Current State-of-the-Art:**
- Metric: num_points
- Best Known Construction Size: 40
- Direction: MAXIMIZE (higher is better)
- Source: Cohn & Rajagopal (2024) 'Variations on five-dimensional sphere packings'
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
# Return unit vectors in R^5 with pairwise dot products <= 1/2 (equivalently, distances >= 1)
return {
"points": [
[x_1, x_2, x_3, x_4, x_5], # Point 1 (must have norm 1)
[x_1, x_2, x_3, x_4, x_5], # Point 2 (must have norm 1)
# ... more points, each in R^5 on the unit sphere
]
}
# Alternative: return just the list of points [[...], [...], ...]
```
**To beat the baseline:** Your result must be > 40
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2412.00937 | Cohn & Rajagopal (2024) 'Variations on five-dimensional sphere packings' - analyzes kissing configurations achieving the bound of 40 in dimension 5, presents fourth known construction | null | null | null | null | null |
kissing_number_dim9 | Consider the following optimization problem.
**Kissing Number in Dimension 9**
**Definition:** The kissing number in 9 dimensions has bounds $306 \le \tau_9 \le 363$. The gap is significant.
**Task:** Construct a valid kissing configuration in $\mathbb{R}^9$ with strictly more than 306 spheres.
**Current State-of-the-Art:**
- Metric: num_points
- Best Known Value: 306 <= k <= 363
- Direction: MAXIMIZE (higher is better)
- Source: High accuracy semidefinite programming bounds for kissing numbers (2010) by Hans D. Mittelmann and Frank Vallentin
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
# Return unit vectors in R^9 with pairwise dot products <= 1/2 (equivalently, distances >= 1)
return {
"points": [
[x_1, x_2, ..., x_9], # Point 1 (must have norm 1)
[x_1, x_2, ..., x_9], # Point 2 (must have norm 1)
# ... more points, each in R^9 on the unit sphere
]
}
# Alternative: return just the list of points [[...], [...], ...]
```
**To beat the baseline:** Your result must be 306 < k < 363
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2412.00937 | Cohn & Rajagopal (2024) 'Variations on five-dimensional sphere packings' - also constructs new kissing configuration in dimension 9 | null | null | null | null | null |
kissing_number_dim11 | Consider the following optimization problem.
**Kissing Number in Dimension 11**
**Definition:** The kissing number in 11 dimensions has bounds $593 \le \tau_{11} \le 868$.
**Task:** Construct a valid kissing configuration in $\mathbb{R}^{11}$ with strictly more than 593 spheres.
**Current State-of-the-Art:**
- Metric: num_points
- Best Known Value: 593
- Direction: MAXIMIZE (higher is better)
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
# Return unit vectors in R^11 with pairwise dot products <= 1/2 (equivalently, distances >= 1)
return {
"points": [
[x_1, x_2, ..., x_11], # Point 1 (must have norm 1)
[x_1, x_2, ..., x_11], # Point 2 (must have norm 1)
# ... more points, each in R^11 on the unit sphere
]
}
# Alternative: return just the list of points [[...], [...], ...]
```
**To beat the baseline:** Your result must be > 593
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/1507.03631 | Novikov et al. (2025) 'AlphaEvolve: A coding agent for scientific and algorithmic discovery' - Improves the lower bound to 593 | null | null | null | null | null |
kakeya_finite_field | Consider the following optimization problem.
**Smaller Kakeya Set in $\mathbb{F}_p^3$**
**Definition:** A Kakeya set in $\mathbb{F}_p^d$ contains a line in every direction. For $d=3$ and primes $p \equiv 1 \pmod 4$, the current best construction has size approx $p^3/4 + 7p^2/8$.
**Task:** Construct an explicit Kakeya set in $\mathbb{F}_p^3$ with cardinality strictly less than the current best known formula.
**Current State-of-the-Art:**
- Metric: Cardinality of a Kakeya set in F_p^3 for p = 1 (mod 4)
- Best Known Value: 0.2107q^3
- Direction: MINIMIZE (lower is better)
- Source: Finite Field Kakeya and Nikodym Sets in Three Dimensions (2018) by Lund et al.
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
# Return a Kakeya set in F_p^3 (must contain a line in every direction)
return {
"p": prime, # The prime defining the finite field F_p
"points": [
[x, y, z], # Point in F_p^3 (coordinates mod p)
[x, y, z],
# ... all points in the Kakeya set
]
}
```
**To beat the baseline:** Your result must be < 0.2107q^3
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/1609.01048 | Lund, Saraf & Wolf (2016) 'Finite field Kakeya and Nikodym sets in three dimensions' - improved lower bounds on Kakeya sets over F_q^3 | null | null | null | null | null |
spherical_9_design_s2 | Consider the following optimization problem.
**Minimal Spherical 9-Design on $S^2$**
**Definition:** A spherical 9-design on the 2-sphere ($S^2 \subset \mathbb{R}^3$) is a finite set of points such that the average of any polynomial of degree $\le 9$ over the points equals the average value over the sphere. The Delsarte-Goethals-Seidel lower bound gives $N \ge 30$ points for a 9-design on $S^2$. The best known construction has 48 points (Hardin & Sloane, 1996). The gap between 30 and 48 is open.
**Task:** Construct a spherical 9-design on $S^2$ with strictly fewer than 48 points.
**Current State-of-the-Art:**
- Metric: num_points
- Best Known Value: 48
- Direction: MINIMIZE (lower is better)
- Source: Hardin & Sloane 1996 'McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions'
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
return {
"points": [
[x1, x2, x3], # Point on S^2 (unit sphere in R^3, norm = 1)
...
]
}
# Alternative: return [[x1, x2, x3], ...] directly
```
Each point must be a 3D vector on the unit sphere (norm = 1).
**To beat the baseline:** Your result must be fewer than 48 valid design points.
| construction | discrete_geometry | benchmark_best_known | 2 | null | https://arxiv.org/abs/math/0207211 | Hardin & Sloane (1996) 'McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions' - provides spherical t-design constructions on S^2, including a 48-point 9-design. The DGS lower bound is 30 points. | null | null | null | null | null |
spherical_7_design_minimal | Consider the following optimization problem.
**Spherical 7-Design with Minimal Points**
**Definition:** Construct a spherical $t$-design for $t=7$ on $S^3$ (dimension 4) with the minimum possible number of points.
**Task:** Construct an explicit spherical 7-design in dimension 4 with fewer points than the current best known construction.
**Current State-of-the-Art:**
- Metric: num_points
- Best Known Value: 48
- Direction: MINIMIZE (lower is better)
- Source: Spherical Designs in Four Dimensions (2004) by Hardin, Sloane, and Cara, Table 1.
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
return {
"points": [[x1, x2, x3, x4], ...] # Points on S^3 (4D unit sphere)
}
# Alternative: return [[x1, x2, x3, x4], ...] directly
```
Each point must be a 4D vector on the unit sphere (norm = 1).
**To beat the baseline:** Your result must be < 48 (the DGS lower bound is 40).
| construction | discrete_geometry | benchmark_best_known | 2 | null | https://www.researchgate.net/publication/4021411_Spherical_designs_in_four_dimensions | Hardin, Sloane, and Cara (2004), 'Spherical Designs in Four Dimensions', Table 1. The best known spherical 7-design on S^3 (4D) uses 48 points. The DGS lower bound is 40 points. | null | null | null | null | null |
keich_thin_triangles_128 | ### Thin-Triangle Kakeya (128 slopes): Minimize Union Area
This benchmark is a *discrete, thickened* Kakeya-type construction in the style of Schoenberg/Keich.
Fix N=128 and δ = 1/128.
For each i=0,1,...,127 you must specify a unit line segment
l_i = {(x, a_i x + b_i) : x in [0,1]} with slope a_i = i/128.
From each segment l_i we define the thin triangle R_δ(l_i) as follows:
- The **upper edge** is l_i.
- The **lower edge** is the segment from (0, b_i - δ) to (1, a_i + b_i).
- The **vertical edge** closes the triangle at x=0.
Equivalently, for x in [0,1], the vertical cross-section of R_δ(l_i) is the interval
y in [a_i x + b_i - δ(1-x), a_i x + b_i].
Your output defines the set
E = union_{i=0}^{127} R_δ(l_i)
and the **score** is Area(E).
### Goal
MINIMIZE Area(E).
### Baseline: AlphaEvolve (Google DeepMind, 2025) found a construction with union area $\approx 0.11481$, improving on Keich's earlier construction ($\approx 0.11921$). This value is not known to be optimal; beating it would constitute progress.
### Required output format
Return a JSON-serializable dict:
```python
def proposed_solution():
# Must output b_i for each slope i/128.
return {
"intercepts": [b_0, b_1, ..., b_127]
}
```
### Constraints enforced
- Exactly 128 intercepts.
- Slopes are fixed to a_i=i/128.
- The validator computes Area(E) by exact piecewise-linear integration of union cross-sections (deterministic).
### To beat the baseline
your construction needs to be valid and achieve area strictly less than $0.1148103258186177$
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2506.13131 | Baseline from AlphaEvolve (Google DeepMind, 2025). The AlphaEvolve triangles conv{(x_i, 0), (x_i + i/128, 0), (x_i + (i+1)/128, 1)} map exactly to our triangles conv{(0, b_i - 1/128), (0, b_i), (1, b_i + i/128)} by swapping coordinates (x, y) → (y, x) and setting b_i = x_i + i/128, an area-preserving transformation. | null | null | null | null | null |
tammes_n15 | Consider the following optimization problem.
**Tammes Problem for $n=15$**
**Definition:** The Tammes problem asks to maximize the minimum distance between any pair of $n$ points on a sphere. For $n=15$, the optimal configuration is not rigorously proven.
**Task:** Construct a configuration of 15 points on $S^2$ achieving a minimum separation distance strictly greater than the current best known, or provide the exact optimal construction.
**Current State-of-the-Art:**
- Metric: minimum angular distance between any pair of points (in degrees)
- Best Known Value: 53.657...
- Direction: MAXIMIZE (higher is better)
- Source: Cohn et al., Spherical Codes database (https://cohn.mit.edu/spherical-codes/)
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
return {
"points": [[x, y, z], ...] # 15 points on S^2 (3D unit sphere)
}
# Alternative: return [[x, y, z], ...] directly
```
Each point must be a 3D vector on the unit sphere (norm = 1). Exactly 15 points required.
**To beat the baseline:** Your result must be > 53.657...
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://cohn.mit.edu/spherical-codes/ | Cohn et al., Spherical Codes database. Best known configuration for n=15 on S^2 has cosine of minimal angle 0.59260590292507377809642492233276 (minimal polynomial 13x^5 - x^4 + 6x^3 + 2x^2 - 3x - 1). Not proven optimal. | null | null | null | null | null |
heilbronn_n12 | Consider the following optimization problem.
**Heilbronn Configuration for $n=12$**
**Definition:** Place $n$ points in a unit square to maximize the minimum area of any triangle formed by three of the points. For $n=12$, the exact optimal value and configuration are unknown.
**Task:** Construct a configuration of 12 points in the unit square where the minimum triangle area is strictly greater than the current best known record.
**Current State-of-the-Art:**
- Metric: minimum area of any triangle formed by three of the points
- Best Known Value: 0.032599
- Direction: MAXIMIZE (higher is better)
- Source: New Lower Bounds for Heilbronn Numbers (2002) by Francesc Comellas and J. Luis A. Yebra
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
return {
"points": [[x, y], ...] # 12 points in [0,1]^2
}
# Alternative: return [[x, y], ...] directly
```
Each point must be a 2D coordinate in the unit square [0,1] x [0,1]. Exactly 12 points required.
**To beat the baseline:** Your result must be > 0.032599
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://www.combinatorics.org/ojs/index.php/eljc/article/view/v9i1r6/pdf | Baseline lower bound for the unit square Heilbronn number at n=12 from Comellas & Yebra (2002): explicit 12-point configuration with minimum triangle area ≈ 0.032599 (rounded). This is a best-known published construction, not a proven optimum. Global-optimization context: Monji, Modir, Kocuk (arXiv:2512.14505) certifies optima for n<=9 and discusses larger n. | null | null | null | null | null |
dts_7_5_min_scope | Consider the following optimization problem.
**Minimum-Scope Difference Triangle Set (7,5)**
An (n,k)-DTS is an nx(k+1) array A with entries a[i][j] such that each row is strictly increasing and normalized:
0 = a[i][0] < a[i][1] < ... < a[i][k]
Define the set of positive within-row differences:
D = { a[i][j] - a[i][j'] : for all i, and 0 <= j' < j <= k }
Validity constraint: all elements of D must be distinct (no repeated differences across any rows).
Scope (objective): m(A) = max_{i,j} a[i][j].
**Metric:** scope
**Direction:** MINIMIZE
**Current State-of-the-Art (best-known in literature):**
- Best known upper bound: m(7,5) <= 112
- Source: Shehadeh-Kingsford-Kschischang (2026), which improves previous best of 113
**To beat the baseline:** output any valid (7,5)-DTS with scope <= 111.
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
return {
"n": 7,
"k": 5,
"rows": [
[0, a01, a02, a03, a04, a05],
[0, a11, a12, a13, a14, a15],
[0, a21, a22, a23, a24, a25],
[0, a31, a32, a33, a34, a35],
[0, a41, a42, a43, a44, a45],
[0, a51, a52, a53, a54, a55],
[0, a61, a62, a63, a64, a65]
]
}
```
All values must be integers and each row must be strictly increasing.
The validator will compute the scope and verify the DTS constraint exactly.
| construction | combinatorics | benchmark_best_known | 1 | null | https://doi.org/10.1002/jcd.22009 | Shehadeh, M., Kingsford, W., & Kschischang, F. R. (2026). 'New Difference Triangle Sets by a Field-Programmable Gate Array-Based Search Technique.' Journal of Combinatorial Designs, 34(1), 37-50. https://doi.org/10.1002/jcd.22009, Table I reports best-known upper bound m(7,5) ≤ 112. | null | scope | minimize | null | Baseline is best-known (not proven optimal): m(7,5) <= 112 reported in Table I of Shehadeh-Kingsford-Kschischang (2025). |
kissing_number_dim6 | Consider the following optimization problem.
**Kissing Number in Dimension 6**
**Definition:** The kissing number $\tau_6$ is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in 6 dimensions. The known bounds are $72 \le \tau_6 \le 77$. The lower bound is achieved by the $E_6$ root system (Korkine & Zolotareff, 1873) and the upper bound was recently improved to 77 by de Laat, Leijenhorst & de Muinck Keizer (2024).
**Task:** Construct a valid kissing configuration in $\mathbb{R}^6$ with strictly more than 72 spheres.
**Current State-of-the-Art:**
- Metric: number_of_spheres
- Best Known Value: 72 <= k <= 77
- Direction: MAXIMIZE (higher is better)
- Source: Korkine & Zolotareff (1873) for lower bound; de Laat, Leijenhorst & de Muinck Keizer (2024) for upper bound
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
# Return points on the unit sphere in R^6 with pairwise distances >= 1
# (equivalently, pairwise dot products <= 1/2)
return {
"points": [
[x_1, x_2, x_3, x_4, x_5, x_6], # Point 1 (must have norm 1)
[x_1, x_2, x_3, x_4, x_5, x_6], # Point 2 (must have norm 1)
# ... more points, each in R^6 on the unit sphere
]
}
# Alternative: return just the list of points [[...], [...], ...]
```
**To beat the baseline:** Your result must be > 72
| construction | discrete_geometry | benchmark_best_known | 1 | null | https://arxiv.org/abs/2404.18794 | D. de Laat, N. Leijenhorst, and W. H. H. de Muinck Keizer, 'Optimality and uniqueness of the D4 root system' (2024). Proves upper bound tau_6 <= 77 via exact semidefinite programming. Lower bound of 72 from E6 root system due to A. Korkine and G. Zolotareff (1873). | null | null | null | null | null |
knot_volume_7_2 | Consider the following research problem in mathematics.
**Hyperbolic Volume of the $7_2$ Knot**
**Definition:** The complement of the knot $7_2$ in the 3-sphere is a hyperbolic 3-manifold with a finite volume (approximately $3.3317...\dots$). The volume is known to be expressible as a sum of Bloch\u2013Wigner dilogarithms at algebraic arguments derived from the ideal triangulation gluing equations (computed routinely by SnapPy/SnapPea). What remains open is whether this volume admits a simpler closed-form expression in terms of standard mathematical constants such as $\pi$, $\zeta(n)$, or $L$-values of number fields.
**Task:** Find a symbolic closed-form expression for the hyperbolic volume of the $7_2$ knot complement in terms of standard mathematical constants (beyond the known Bloch\u2013Wigner dilogarithm representation).
Propose your best conjectured solution, which will be evaluated extensively against high-precision numerical ground-truth solutions. Even if your solution is unproven, you must propose a candidate solution (like your own novel conjecture) that satisfies the requirements listed above. REQUIRED OUTPUT FORMAT:
A Python function using mpmath that computes your expression with the following structure.
def proposed_solution():
from mpmath import mp
mp.dps = 100 # decimal places of precision
# Use only mpmath functions: constants (mp.pi, mp.e, mp.euler, mp.catalan),
# elementary functions (mp.sqrt, mp.exp, mp.log, ...),
# special functions (mp.gamma, mp.zeta, mp.ellipk, mp.hyper, ...)
# No other functions are permitted. **For this problem, the known Bloch-Wigner dilogarithm representation is explicitly forbidden.**
result = ... # your closed-form expression
return result | constant | discrete_geometry | ground_truth_computable | 2 | 3.3317442316411148239145691080297127955469579091860049212216044555987413728423665155788622603487862838857647164 | https://katlas.org/wiki/7_2 | Knot Atlas 7_2 page gives 3.33174, and Wakelin (2023)'s 'A hyperbolic perspective on the Dehn surgery characterisation problem' lists 3.3317442316. | null | null | null | null | null |
diff_basis_upper | Consider the following optimization problem.\n\n**Improve Upper Bound on Difference Basis Constant**\n\n**Definition:** For any natural number $n$, let $\Delta(n)$ denote the size of the smallest set $B$ of integers such that every natural number $k \in \{1,\dots,n\}$ is expressible as a difference of two elements of $B$ (equivalently, for each $k$ there exist $a,b \in B$ with $|a-b| = k$). Define $C_{6.7}(n) := \Delta(n)^2/n$, and $C^{6.7} := \inf_{n \ge 1} C_{6.7}(n)$. The constant $C^{6.7}$ has a verified upper bound of $2.6390$.\n\n**Task:** Construct a family of difference bases that establishes an upper bound on $C^{6.7}$ strictly less than $2.6390$.\n\n**Current State-of-the-Art:**\n- Metric: Upper bound on the constant $C^{6.7} = \inf_{n \ge 1} \Delta(n)^2/n$ (certified by $\text{ratio} = |B|^2/n$ for a valid difference basis $B$ for $\{1,\dots,n\}$)\n- Best Known Value: $2.6390$\n- Direction: MINIMIZE (lower is better)\n- Source: Mathematical exploration and discovery at scale (2025) by Bogdan Georgiev et al.\n\n**REQUIRED OUTPUT FORMAT:**\n\ndef proposed_solution():\n # Your solution code here\n return {\n \"n\": n_value, # the interval {1, ..., n}\n \"basis\": [b0, b1, b2, ...] # a list of integers (may be negative)\n }\n\n**To beat the baseline:** Your result must be $< 2.6390$.\n\n | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2103.15850 | Balogh, Furedi & Roy (2021) 'An upper bound on the size of Sidon sets' - proves maximum Sidon set size is at most sqrt(n) + 0.998n^(1/4), directly related to difference basis bounds | null | null | null | null | null |
diff_basis_optimal_10000 | Consider the following optimization problem.
**Restricted Difference Basis (Sparse Ruler) for n=10000**
**Definition:** A set B ⊆ {0,1,...,9999} is a restricted difference basis for n=10000 if every integer d in {1,...,9999} can be written as |a-b| for some a,b ∈ B.
**Task:** Construct an explicit B with |B| smaller than the current best-known construction.
**Current Best-Known Construction (Upper Bound):**
- Metric: basis_size = |B|
- Best Known Value (construction): 174 (from the known constructive bound via sparse-ruler/excess-0/1 constructions)
- Direction: MINIMIZE (lower is better)
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
return {"basis": [b0, b1, b2, ...]} # integers in [0, 9999]
# or simply: return [b0, b1, b2, ...]
**To beat the baseline:** Your result must have |B| < 174.
| construction | combinatorics | benchmark_best_known | 2 | null | https://oeis.org/A046693 | Sparse ruler / minimal complete ruler context; excess discussion also in OEIS A326499 and Wolfram references. | null | null | null | null | null |
vdw_W72_ap7 | Consider the following optimization problem.
**2-Coloring with No Monochromatic 7-Term Arithmetic Progression**
**Definition (certificate format):** A candidate solution is a list `c[0..n-1]` with entries in {0,1}, interpreted as a 2-coloring of the integers {0,1,...,n-1}.
A **7-term arithmetic progression** in {0,...,n-1} is any set of the form:
a, a+d, a+2d, a+3d, a+4d, a+5d, a+6d
where d is a positive integer and a+6d <= n-1.
The coloring is **valid** if **no** such 7-term progression is monochromatic (i.e., not all 7 values are equal).
**Task:** Output a valid coloring and maximize `n`.
**Metric:** length (n)
**Direction:** MAXIMIZE (higher is better)
**Current Best Known Baseline (published lower bound):** n = 3703
(Equivalently: W(7,2) > 3703, so a valid 2-coloring of length 3703 exists.)
**To beat the baseline:** Your coloring must be valid and have n >= 3704.
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Return either a list of 0/1 values, or a dict with key "coloring".
# Example:
# return {"coloring": [0,1,0,1,...]}
return {"coloring": [...]}
```
**Notes:**
- Your output may be any length n >= 1.
- Entries must be exactly 0 or 1 (integers).
- The validator checks all 7-term arithmetic progressions with d>0 within the range.
| construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/1603.03301 | Monroe (2019) compiles lower bounds from explicit constructions; reports W(7,2) > 3703 (baseline). | null | null | null | null | null |
general_diff_basis_algo | Consider the following optimization problem.
**General Algorithm for Difference Bases**
**Definition:** Construct a deterministic algorithm or formula that generates difference bases for any range $n$ with size close to the theoretical lower bound, replacing sporadic search-based results.
**Task:** Find a universal algorithm or formula $B(n)$ that produces a valid difference basis for range $n$ with efficiency $|\Delta(n)|^2/n$ consistently lower than current general methods.
**Current State-of-the-Art:**
- Metric: efficiency |Delta(n)|^2/n
- Best Known Value: (2 * ceil(sqrt(n)))^2 / n
- Direction: MINIMIZE (lower is better)
- Source: Cardinalities of g-difference sets (2025) by Eric Schmutz and Michael Tait
**REQUIRED OUTPUT FORMAT:**
Your solution must be returned as a Python dictionary.
def proposed_solution():
# Your solution code here
return {
"algorithm": "description",
"test_cases": [
{"n": n, "basis": [b0, b1, ...]},
# ... more test cases
]
}
**To beat the baseline:** Your result must be < (2 * ceil(sqrt(n)))^2 / n
| formula_discovery | combinatorics | benchmark_best_known | 2 | null | https://en.wikipedia.org/wiki/Difference_set | Wikipedia article on difference sets. Singer (1938) proved perfect difference sets exist mod (q^2+q+1) when q is prime power. General algorithmic construction for difference bases not found in verified sources. | null | null | null | null | null |
ramsey_asymptotic | Consider the following optimization problem.
**Asymptotic Upper Bound Constant for Diagonal Ramsey Numbers**
**Definition:** The diagonal Ramsey numbers satisfy classical bounds of the form $2^{n/2} \lesssim R(n,n) \lesssim 4^n$.
**Goal:** Improve the best known exponential **upper bound base** $c$ in $R(k,k) \le c^{k+o(k)}$.
**Current State-of-the-Art:**
- Metric: Upper bound base $c$ in $R(k,k) \le c^{k+o(k)}$
- Best Known Value: $c \approx 3.7992...$
- Direction: MINIMIZE (lower $c$ is better)
- Source: Gupta, Ndiaye, Norin, Wei (2024), "Optimizing the CGMS upper bound on Ramsey numbers"
---
### Mathematical framework
Gupta-Ndiaye-Norin-Wei (2024) prove that $R(k,\ell) \le e^{F(\ell/k)\,k + o(k)}$ provided the following sufficient conditions hold for all $\lambda \in (0,1]$.
Let $F:(0,1] \to \mathbb{R}_+$ be smooth, and let $M, Y:(0,1] \to (0,1)$. Define
$$X(\lambda) = \bigl(1 - e^{-F'(\lambda)}\bigr)^{1/(1-M(\lambda))}\,(1 - M(\lambda)).$$
The sufficient conditions are:
1. $F(\lambda) > 0$, $\;F'(\lambda) > 0$
2. $(X(\lambda),\, Y(\lambda)) \in \mathcal{R}$, the admissible Ramsey region
3. $F(\lambda) > -\tfrac{1}{2}\bigl(\log X(\lambda) + \lambda\log M(\lambda) + \lambda\log Y(\lambda)\bigr)$
The resulting diagonal bound is $c = e^{F(1)}$.
For this problem, $F$ is parameterized as
$$F(\lambda) = (1+\lambda)\log(1+\lambda) - \lambda\log\lambda + p(\lambda)\,e^{-\lambda},$$
where $p(\lambda)$ is a polynomial in $\lambda$ with **no constant term**:
$$p(\lambda) = a_1\lambda + a_2\lambda^2 + \cdots + a_d\lambda^d$$
for some finite degree $d \ge 1$ chosen by the submission.
---
### Split-regime benchmark design
The benchmark uses two different regimes.
#### Small-$\lambda$ regime: $0 < \lambda \le \lambda_0$
Set
$$\lambda_0 = 10^{-3}, \qquad \alpha_{\mathrm{small}} = (0.17 - 0.033)e^{-1}.$$
On this interval, the validator does **not** use the submitted step functions. Instead it uses the fixed analytic choices
$$M(\lambda) = \lambda e^{-\lambda},$$
and
$$Y(\lambda) =
\begin{cases}
e^{\alpha_{\mathrm{small}}}(1 - X(\lambda)), & X(\lambda) \le \tfrac12,\\[1mm]
1 - X(\lambda)e^{-\alpha_{\mathrm{small}}}, & X(\lambda) > \tfrac12.
\end{cases}$$
This part is motivated by Lemma 14 and the iteration used in Theorem 1 of GNNW (2024): the validator treats the admissibility condition $(X(\lambda),Y(\lambda)) \in \mathcal{R}$ on $(0,\lambda_0]$ via that theorem-backed analytic construction.
So in the small-$\lambda$ regime the submission only influences the validator through the polynomial coefficients of $p$, i.e. through $F$.
#### Large-$\lambda$ regime: $[\lambda_0,1]$
On $[\lambda_0,1]$, the submission provides piecewise-constant functions $M$ and $Y$.
Condition (2) is checked against the fixed inner approximation $\mathcal{R}_0 \subseteq \mathcal{R}$:
$$U(\mu) = G(\mu) + (1+\mu)\log(1+\mu) - \mu\log\mu,$$
with
$$G(\mu) = (-0.25\mu + 0.033\mu^2 + 0.08\mu^3)e^{-\mu}.$$
A pair $(x,y) \in \mathcal{R}_0$ iff
$$-\log x - \mu\log y \ge U(\mu) \qquad \text{for all } \mu \in (0,1].$$
Since $R(k,\ell) = R(\ell,k)$, the pair $(x,y)$ is accepted if either $(x,y) \in \mathcal{R}_0$ or $(y,x) \in \mathcal{R}_0$.
**To beat the baseline**, find parameters giving $e^{F(1)} < 3.7992...$ while satisfying the theorem conditions under the split validator above.
---
### Validation method
The validator uses **rigorous interval arithmetic** (via `mpmath.iv` at 100-digit precision) to certify all conditions on every sub-interval. This means:
- If the validator accepts, the conditions are **provably** satisfied.
- Solutions need sufficient **slack margin** (typically $\ge 10^{-4}$) so that interval widening across sub-intervals does not push certified lower bounds below zero.
- Solutions with very tight margins may be mathematically correct but fail validation.
---
### REQUIRED OUTPUT FORMAT
```python
def proposed_solution():
return {
"polynomial_coeffs": [a1, a2, ..., ad],
"M": {"breakpoints": [b1, b2, ...], "values": [v0, v1, v2, ...]},
"Y": {"breakpoints": [b1, b2, ...], "values": [v0, v1, v2, ...]},
"notes": "..."
}
```
- `polynomial_coeffs`: a list of floats $[a_1, a_2, \ldots, a_d]$ for $p(\lambda) = a_1\lambda + a_2\lambda^2 + \cdots + a_d\lambda^d$. Any degree $d \ge 1$ is allowed.
- `M`, `Y`: piecewise-constant step functions on $[\lambda_0,1]$.
- `breakpoints` must be a strictly increasing list in $(\lambda_0,1)$ with at most 500 elements.
- `values` must have length `len(breakpoints) + 1`, with every value in $(0,1)$.
- The piecewise convention is: `values[0]` is used for $\lambda < $ `breakpoints[0]`, `values[i]` for `breakpoints[i-1]` $\le \lambda < $ `breakpoints[i]`, and `values[-1]` for $\lambda \ge $ `breakpoints[-1]`.
- `notes` may contain any brief explanation.
The validator ignores any breakpoints $\le \lambda_0 = 10^{-3}$ and their associated values.
Return the dictionary. | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2407.19026 | Gupta, Ndiaye, Norin, Wei (2024) 'Optimizing the CGMS upper bound on Ramsey numbers'. Baseline c = 4·exp(-0.14/e) = 3.7992… from Theorem 1. Arbitrary-degree polynomial correction p(λ) = a1·λ + … + ad·λ^d (no constant term). Split validator with rigorous interval arithmetic: on (0, 10^-3] it uses fixed analytic M(λ)=λe^{-λ} and Y(λ) from Lemma 14/Theorem 1. On [10^-3,1], submitted piecewise-constant M and Y are checked against the symmetric inner approximation R0 (β=0.033). Solutions need sufficient slack margin (~1e-4) to survive interval widening. | null | null | null | null | null |
crossing_number_kn | Consider the following optimization problem.
**Rectilinear Crossing Number $\overline{\mathrm{cr}}(K_n)$ (Straight-Line Drawings)**
**Definition:** A *rectilinear drawing* of the complete graph $K_n$ is obtained by placing $n$ points in the plane in general position (no three collinear) and drawing each edge as the straight-line segment between its endpoints. The *rectilinear crossing number* $\overline{\mathrm{cr}}(K_n)$ is the minimum possible number of edge crossings over all such rectilinear drawings.
**Task:** Construct an explicit rectilinear drawing of $K_{99}$ (as vertex placements in the plane) that minimizes the number of edge crossings. The validator counts crossings in your straight-line drawing.
**Current State-of-the-Art Baseline (published upper bound):**
- Metric: crossing_count (number of crossings in your drawing of $K_{99}$)
- Baseline Value: 1404552
- Direction: MINIMIZE (lower is better)
- Source: Ábrego et al., *3-symmetric and 3-decomposable geometric drawings of $K_n$* (extended version), construction of a drawing of $K_{99}$ with 1404552 crossings.
**REQUIRED OUTPUT FORMAT:**
Your solution must be returned as a Python dictionary.
def proposed_solution():
# Construct vertex placements for K_99
# Points must be in general position (no 3 collinear)
return {
"drawings": [
{
"n": 99,
"points": [[x1, y1], [x2, y2], ...] # exactly 99 points
}
]
}
**To beat the baseline:** Achieve crossing_count < 1404552.
**Notes:**
- All points must be distinct and in general position (no three collinear).
- The validator counts crossings between all pairs of non-adjacent edges in the straight-line drawing.
- Coordinates must be finite real numbers (no NaN/Inf). | formula_discovery | combinatorics | benchmark_best_known | 2 | null | https://www.sciencedirect.com/science/article/pii/S0166218X09003734 | Baseline is an explicit published rectilinear drawing of K_99 with 1404552 crossings (Ábrego et al. (2010). 'How to construct a drawing of K_99 with 1404552 crossings'). | null | null | null | null | null |
kcore_threshold_c3 | Consider the following research problem in mathematics.\n\n**3-Core Emergence Threshold Constant in G(n, c/n)**\n\n**Definition:** Let G(n,p) be the Erd\u0151s\u2013R\u00e9nyi random graph. The 3-core of a graph is its largest induced subgraph with minimum degree at least 3. There exists a sharp threshold at p = c_3/n for the appearance of a non-empty (indeed linear-size) 3-core as n\u2192\u221e.\n\nDefine \u03c0_3(\u03bb) = P(Poisson(\u03bb) \u2265 2) = 1 - e^{-\u03bb}(1+\u03bb). The threshold constant is\n\n c_3 := min_{\u03bb>0} \u03bb / \u03c0_3(\u03bb).\n\n(Equivalently, if \u03bb_* is the unique root in (1,2) of e^{\u03bb} = 1 + \u03bb + \u03bb^2, then c_3 = \u03bb_* + 1 + 1/\u03bb_*.)\n\n**Task:** Find a closed-form expression for c_3 that matches the high-precision numeric target.\n\nREQUIRED OUTPUT FORMAT:\n\nA Python function using mpmath that computes your expression with the following structure.\n\n\ndef proposed_solution():\n from mpmath import mp\n mp.dps = 100\n # Use only mpmath constants, elementary functions, and special functions.\n # Forbidden: numerical root-finding (findroot), numerical quadrature (quad), nsum/nprod, truncated approximation loops, or hardcoding long decimal expansions.\n result = ...\n return result\n | constant | mathematical_constants | ground_truth_computable | 2 | 3.35091887151167277315681440498709807619062659090935600532811122807017749104521799074756363155452191680828276744801164941414782014826348832037202660117572096525917495822458142281358203481658555212080736970109895 | https://cs.nyu.edu/~spencer/papers/k-core.pdf | Pittel, Spencer, Wormald (1996) define the k-core threshold for G(n,m) as c_k = min_{\u03bb>0} \u03bb/\u03c0_k(\u03bb), with \u03c0_k(\u03bb)=P(Poisson(\u03bb)\u2265 k-1), and state c_3\u22483.35. Later work quotes the more precise value qc\u22483.35091887 for k=3 (e.g. Baxter et al., Phys. Rev. X 5, 031017 (2015)). | null | null | null | null | null |
turan_petersen | Consider the following optimization problem.
**Petersen Graph Tur\'an Problem**
**Definition:** Find the maximum number of edges in a graph on $n=50$ vertices that does not contain the Petersen graph as a subgraph.
**Task:** Construct an explicit graph on 50 vertices with no Petersen subgraph achieving a higher edge count than current best bounds.
**Current State-of-the-Art:**
- Metric: number_of_edges
- Best Known Value: 673
- Direction: MAXIMIZE (higher is better)
- Source: Simonovits-type extremal construction family $H(n,2,3)=K_2\nabla T_2(n-2)$; for $n=50$, this gives $K_2\nabla K_{24,24}$ with $576+96+1=673$ edges.
**REQUIRED OUTPUT FORMAT:**
Your solution must be returned as a Python dictionary.
def proposed_solution():
import numpy as np
# Your solution code here
return {
"n": number_of_vertices,
"edges": [[u, v], ...] # 0-indexed vertex pairs
} | construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/pdf/2508.12070 | Fang, Lin, Zhai (2025), 'The spectral Turan problem: Characterizing spectral-consistent graphs.' For n=50: T_2(48)=K_{24,24} has 24*24=576 edges, and joining two universal vertices adds 2*48=96 edges, plus the edge between them adds 1. Total is 576+96+1=673. | null | null | null | null | null |
ramsey_coloring_k5 | Consider the following optimization problem.
**2-Coloring of $K_n$ Without Monochromatic $K_5$**
**Definition:** The Ramsey number $R(5,5)$ is unknown (bounds: 43-48). Constructing a coloring for a specific $n$ (e.g., $n=43$) without a monochromatic $K_5$ would improve the lower bound.
**Task:** Construct an explicit 2-coloring of the edges of $K_{43}$ (or higher $n$) that contains no monochromatic clique of size 5.
**Current State-of-the-Art:**
- Metric: lower bound for Ramsey number R(5,5)
- Best Known Value: 43
- Direction: MAXIMIZE (higher is better)
- Source: Study of Exoo's Lower Bound for Ramsey Number R(5,5) (2023) by Lachlan Ge et al.
**REQUIRED OUTPUT FORMAT:**
Return your solution as a Python function that returns a dictionary:
def proposed_solution():
# Your solution code here
return {
"n": int, # number of vertices (must be > 43 to beat baseline)
"coloring": [[int, ...], ...] # n x n symmetric matrix with entries 0 or 1
}
# Alternative format:
# return {
# "n": int,
# "red_edges": [[u, v], ...], # edges of color 0 (red)
# "blue_edges": [[u, v], ...] # edges of color 1 (blue), optional
# }
**To beat the baseline:** Your result must be > 43
| construction | combinatorics | benchmark_best_known | 1 | null | https://arxiv.org/abs/2212.12630 | Study of Exoo's lower bound for R(5,5) - analyzes the 2-coloring of K_42 with no monochromatic K_5, establishing R(5,5) >= 43. Current bounds: 43 <= R(5,5) <= 46. | null | null | null | null | null |
merit_factor_6_5 | Consider the following research problem in mathematics.
**Polynomial with Maximum Merit Factor**
**Definition:** The merit factor of a binary polynomial $p(z) = \sum_{i=0}^{n-1} a_i z^i$ with coefficients $a_i \in \{-1, 1\}$ is:
$$F(p) = \frac{n^2}{2 \sum_{k=1}^{n-1} C_k^2}$$
where $C_k = \sum_{i=0}^{n-1-k} a_i a_{i+k}$ is the aperiodic autocorrelation at lag $k$.
The merit factor measures how flat the polynomial's magnitude is on the unit circle. The best known constructions achieve a merit factor of approximately $9.5851$ (Borwein et al., 2004).
**Task:** Construct a binary polynomial of length $n \geq 100$ with coefficients in $\{-1, 1\}$ whose merit factor is strictly greater than $9.5851.
**Constraints:**
- All coefficients must be in $\{-1, 1\}$
- The sequence length must be at least $100$ and the merit factor must exceed $9.5851$
**Current State-of-the-Art:**
- Metric: merit factor
- Best Known Value: $F=9.5851$ for $L=191,E=1903$.
- Direction: MAXIMIZE (higher is better)
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution():
# Your solution code here
return {"coefficients": [a_0, a_1, ..., a_n_minus_1]} # where each a_i is in {-1, 1} and n >= 100
# or simply: return [a_0, a_1, ..., a_n_minus_1]
``` | construction | coding_theory | benchmark_best_known | 2 | null | https://ieeexplore.ieee.org/document/8247176/ | Brest, J., & Bošković, B. (2018). A heuristic algorithm for a low autocorrelation binary sequence problem with odd length and high merit factor. IEEE Access, 6, 4127-4134. | null | null | null | null | null |
parametric_spherical_codes | Consider the following optimization problem.
**Parametric Family of Spherical Codes**
**Definition:** Discover a parametric family of spherical codes (depending on dimension $d$ and size $N$) that produces configurations with high minimum distance, generalizing isolated optimal codes.
**Task:** Find a universal formula or construction rule for spherical codes that yields near-optimal minimum distances for a broad class of $(d, N)$ parameters.
**Current State-of-the-Art:**
- Metric: cardinality (number of codewords) for a given minimum Euclidean distance
- Best Known Value: N = 2^(4k) + 2^(2k+1) points in n = 2^(2k) dimensions (for 2 <= k <= 5) with corresponding minimal angle θ. For n=16, N=288, cos θ = 1/4.
- Direction: MAXIMIZE (higher is better)
- Source: Optimality of Spherical Codes via Exact Semidefinite Programming Bounds (2024) by Henry Cohn et al.
**REQUIRED OUTPUT FORMAT:**
Your solution must be returned as a Python dictionary.
def proposed_solution():
import numpy as np
# Your solution code here
return {
"family": "description",
"test_cases": [
{"n": num_points, "dimension": d, "points": [[...], ...], "min_distance": dist},
# ... more test cases
]
}
**To beat the baseline:** Your result must be > N = 2^(4k) + 2^(2k+1) points in n = 2^(2k) dimensions (for 2 <= k <= 5) with corresponding minimal angle θ. For n=16, N=288, cos θ = 1/4.
| formula_discovery | coding_theory | benchmark_best_known | 2 | null | https://arxiv.org/abs/2008.10728 | Miyamoto, Costa, Sa Earp, 'Constructive Spherical Codes by Hopf Foliations' (2021). Parametric family construction in dimensions 2^k using Hopf foliations. O(n) storage, O(n log n) encoding. Published in IEEE Trans. Inf. Theory 67(12):7925-7939. | null | null | null | null | null |
bklc_68_15 | Consider the following optimization problem.
**Improve Minimum Distance of a Binary Linear [68,15] Code**
**Definition:** A binary linear [n,k,d] code is a k-dimensional subspace of F_2^n. Its minimum distance d is the minimum Hamming weight among all nonzero codewords.
**Task:** Construct an explicit binary linear code with n=68 and k=15 that maximizes the minimum distance d.
**Certificate format (generator matrix):** Provide a generator matrix G in GF(2) of shape 15 x 68. The code is the set of all 2^15 linear combinations of the rows of G.
**Current State-of-the-Art:**
- Metric: min_distance
- Best Known Value (lower bound): 24
- Best Known Upper Bound: 26
- Direction: MAXIMIZE (higher is better)
- Source: Best Known Linear Codes tables (Markus Grassl), entry for [68,15] over GF(2)
**REQUIRED OUTPUT FORMAT:**
Return your solution as a Python function that returns a dictionary:
def proposed_solution():
return {
"generator_matrix": [
"0101...", # 68-character bitstring
"...", # total of 15 rows
"..."
]
}
Notes:
- Each row must be a string of exactly 68 characters in {0,1}.
- The validator will check that rank(G)=15 over GF(2) and will compute the exact minimum distance by enumerating all 2^15 codewords.
**To beat the baseline:** Your min_distance must be > 24. | construction | coding_theory | benchmark_best_known | 2 | null | https://www.codetables.de/BKLC/BKLC.php?k=15&n=68&q=2 | Grassl BKLC lists lower bound 24 and upper bound 26 for binary linear codes with (n,k)=(68,15), so 24 is best-known but not proven optimal. | null | null | null | null | null |
covering_C13_k7_t4 | Consider the following optimization problem.
**Covering Design $C(13,7,4)$ With Fewer Blocks**
**Definition:** A candidate solution is a list of blocks (each block is a 7-element subset of {0,1,...,12}). The solution is valid if every 4-element subset of {0,1,...,12} is contained in at least one block.
**Task:** Output a covering design for C(13,7,4) with as few blocks as possible.
**Current State-of-the-Art:**
- Metric: num_blocks
- Best Known Value: 30
- Direction: MINIMIZE (lower is better)
- Source: La Jolla Covering Repository reports 28 ≤ C(13,7,4) ≤ 30 and provides an explicit 30-block cover.
**REQUIRED OUTPUT FORMAT:**
Return your solution as a Python function that returns a dictionary:
```python
def proposed_solution():
return {
"blocks": [[int, int, int, int, int, int, int], ...] # each block has length 7
}
```
**To beat the baseline:** your design must be valid and have num_blocks < 30.
| construction | coding_theory | benchmark_best_known | 2 | null | https://ljcr.dmgordon.org/cover/show_cover.php?k=7&t=4&v=13 | Baseline uses LJCR explicit cover for C(13,7,4), currently giving 28 ≤ C(13,7,4) ≤ 30. | null | null | null | null | null |
A21_10_binary_code | Consider the following optimization problem.
**Binary Code A(21,10)**
**Definition:** Let A(n,d) be the maximum possible size of a binary code C \subseteq {0,1}^n such that the Hamming distance between any two distinct codewords is at least d. In this problem, n=21 and d=10.
**Task:** Construct an explicit binary code C \subseteq {0,1}^{21} with minimum Hamming distance at least 10 that has strictly more codewords than the current best-known construction.
**Current State-of-the-Art:**
- Metric: number_of_codewords
- Best Known Value (lower bound): 42
- Direction: MAXIMIZE (higher is better)
- Source for lower bound: Kaikkonen (1989) construction implying A(21,10) >= 42
- Known upper bound (for context): A(21,10) <= 47 (semidefinite programming bound)
**REQUIRED OUTPUT FORMAT:**
Return a Python function `proposed_solution()` that returns a JSON-serializable dict:
```
def proposed_solution():
return {
"codewords": [ ... ]
}
```
`codewords` must be a list of codewords, each either:
- an integer in [0, 2^21), interpreted as a 21-bit vector, OR
- a string of length 21 over characters {0,1}.
**To beat the baseline:** your code must have |C| > 42 and satisfy the distance constraint.
| construction | coding_theory | benchmark_best_known | 3 | null | https://aeb.win.tue.nl/codes/binary-1.html | Lower bound A(21,10) >= 42 attributed to M.K. Kaikkonen (IEEE Trans. Inf. Theory 35 (1989) p. 1344). Upper bound A(21,10) <= 47 given by Gijswijt-Mittelmann-Schrijver via semidefinite programming. | null | null | null | null | null |
cwcode_29_8_5 | Consider the following optimization problem.\n\n**Constant-Weight Code A(29,8,5): Pack Pairs by Quintuples**\n\n**Definition:** Let A(n,d,w) be the maximum size of a binary constant-weight code of length n, weight w, and minimum Hamming distance at least d. Here n=29, w=5, d=8. Equivalently, represent each codeword as a 5-subset (block) of {0,...,28}; the distance constraint d>=8 is equivalent to requiring any two blocks intersect in at most one point (i.e., no pair of points appears in two different blocks). A solution is a collection of blocks, where each block is a 5-element subset of the vertex set {0,1,...,28}. The solution is valid iff no unordered pair {i,j} with i<j appears together in more than one block (equivalently, any two blocks intersect in at most one point).\n\n**Objective:** Maximize the number of blocks.\n\n**Current State-of-the-Art:**\n- Metric: num_blocks\n- Best Known Value (published lower bound): 36\n- Direction: MAXIMIZE (higher is better)\n- Source: Bluskov (Electronic Notes in Discrete Mathematics 65 (2018)) as summarized in Brouwer's constant-weight code tables.\n\n**REQUIRED OUTPUT FORMAT:**\nReturn your solution as a Python function that returns a dictionary:\n\n def proposed_solution():\n return {\n \"blocks\": [[int,int,int,int,int], ...]\n }\n\nEach block must contain 5 distinct integers in [0,28].\n\n**To beat the baseline:** Your result must have num_blocks > 36. | construction | coding_theory | benchmark_best_known | 2 | null | https://aeb.win.tue.nl/codes/Andw.html | Brouwer's table lists A(29,8,5) in the A(n,8,5) section as 36^{Bl}-39 and cites Bluskov (ENDM 65 (2018), 31-36) for the lower bound 36. | null | null | null | null | null |
three_mols_order_10 | **Three Mutually Orthogonal Latin Squares of Order 10**
**Definition:** A Latin square of order $n$ is an $n \times n$ array filled with $n$ different symbols (typically $0, 1, \ldots, n-1$) such that each symbol appears exactly once in each row and exactly once in each column.
Two Latin squares $L_1$ and $L_2$ of the same order are **orthogonal** if, when superimposed, every ordered pair of symbols occurs exactly once. Formally, for all $(a, b) \in \{0,\ldots,n-1\}^2$, there exists exactly one position $(i,j)$ such that $L_1[i,j] = a$ and $L_2[i,j] = b$.
A set of Latin squares is **mutually orthogonal** (MOLS) if every pair in the set is orthogonal. For order $n$, at most $n-1$ MOLS can exist.
**Known results for order 10:**
- At least 2 MOLS of order 10 exist (easily constructed)
- A complete set of 9 MOLS does NOT exist (equivalently, no projective plane of order 10 exists — proven by Lam, Thiel, and Swiercz in 1989 via exhaustive computer search)
- Whether 3 MOLS of order 10 exist is **unknown**
This is one of the most famous open problems in combinatorial design theory, open since the 1960s.
**Task:** Construct three Latin squares $L_1, L_2, L_3$ of order 10 (each a $10 \times 10$ array with entries from $\{0, 1, \ldots, 9\}$) such that every pair $(L_1, L_2)$, $(L_1, L_3)$, and $(L_2, L_3)$ is orthogonal.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Each Li is a 10x10 matrix (list of lists) with entries from 0 to 9
L1 = [[...], ...] # First Latin square
L2 = [[...], ...] # Second Latin square
L3 = [[...], ...] # Third Latin square
return {
"squares": [L1, L2, L3]
}
# Alternative format: return [L1, L2, L3] | construction | coding_theory | new_construction | 1 | null | https://arxiv.org/abs/2103.11018 | Integer and Constraint Programming for MOLS; existence of three MOLS of order 10 remains open problem; estimates running time to resolve. Bright, Keita, and Stevens (2026) also state N(10) is still an open problem in their paper 'Myrvold's Results on Orthogonal Triples of 10 x 10 Latin Squares: A SAT Investigation'. | null | null | null | null | null |
hadamard_668 | **Hadamard Matrix of Order 668 via Goethals-Seidel Construction**
**Definition:** A **Hadamard matrix** of order $n$ is an $n \times n$ matrix $H$ with entries in $\{-1, +1\}$ satisfying:
$$H H^T = n I_n$$
where $I_n$ is the $n \times n$ identity matrix. Equivalently, the rows of $H$ are mutually orthogonal.
**Existence conditions:** Hadamard matrices can only exist for $n = 1, 2$, or $n \equiv 0 \pmod 4$.
**The Hadamard Conjecture (1893):** A Hadamard matrix exists for every order $n$ divisible by 4.
**Known constructions:** Hadamard matrices have been constructed for most orders $n \equiv 0 \pmod 4$ up to 2000, using methods including:
- Sylvester construction (powers of 2)
- Paley construction (using quadratic residues when $n-1$ or $n/2 - 1$ is prime)
- Tensor products of smaller Hadamard matrices
- Turyn-type and Goethals-Seidel constructions
**Open cases:** As of 2024, the smallest order for which no Hadamard matrix is known is **668**. Note that $668 = 4 \times 167$ where $167$ is prime.
**Goethals-Seidel construction:** A Hadamard matrix of order $4n$ can be constructed from four $\{+1, -1\}$ sequences $a, b, c, d$ of length $n$ that define circulant matrices $A, B, C, D$ (each sequence is the first row of its circulant) satisfying:
$$AA^T + BB^T + CC^T + DD^T = 4n \cdot I_n$$
The full Hadamard matrix is then assembled as:
$$H = \begin{pmatrix} A & BR & CR & DR \\ -BR & A & D^TR & -C^TR \\ -CR & -D^TR & A & B^TR \\ -DR & C^TR & -B^TR & A \end{pmatrix}$$
where $R$ is the $n \times n$ reversal (back-identity) matrix with $R_{ij} = \delta_{i+j,\, n-1}$.
**Task:** Find four $\{+1, -1\}$ sequences of length 167 that satisfy the Goethals-Seidel condition above, yielding a Hadamard matrix of order 668.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Four sequences of length 167 with entries +1 or -1
# Each sequence is the first row of a 167x167 circulant matrix
a = [1, -1, 1, ...] # length 167
b = [1, 1, -1, ...] # length 167
c = [-1, 1, 1, ...] # length 167
d = [1, -1, -1, ...] # length 167
return {
"rows": [a, b, c, d]
} | construction | coding_theory | new_construction | 1 | null | https://oeis.org/A007299 | OEIS lists 668 as smallest order with no known Hadamard matrix construction; remains open whether H(668) exists | null | null | null | null | null |
hadamard_716 | **Hadamard Matrix of Order 716 via Goethals-Seidel Construction**
**Definition:** A **Hadamard matrix** of order $n$ is an $n \times n$ matrix $H$ with entries in $\{-1, +1\}$ satisfying:
$$H H^T = n I_n$$
where $I_n$ is the $n \times n$ identity matrix. Equivalently, the rows of $H$ are mutually orthogonal.
**Existence conditions:** Hadamard matrices can only exist for $n = 1, 2$, or $n \equiv 0 \pmod 4$.
**The Hadamard Conjecture (1893):** A Hadamard matrix exists for every order $n$ divisible by 4.
**Known constructions:** Hadamard matrices have been constructed for most orders $n \equiv 0 \pmod 4$ up to 2000, using methods including:
- Sylvester construction (powers of 2)
- Paley construction (using quadratic residues when $n-1$ or $n/2 - 1$ is prime)
- Tensor products of smaller Hadamard matrices
- Turyn-type and Goethals-Seidel constructions
**Open cases:** As of 2025, order **716** remains one of the unresolved Hadamard orders. Note that $716 = 4 \times 179$ where $179$ is prime.
**Goethals-Seidel construction:** A Hadamard matrix of order $4n$ can be constructed from four $\{+1, -1\}$ sequences $a, b, c, d$ of length $n$ that define circulant matrices $A, B, C, D$ (each sequence is the first row of its circulant) satisfying:
$$AA^T + BB^T + CC^T + DD^T = 4n \cdot I_n$$
The full Hadamard matrix is then assembled as:
$$H = \begin{pmatrix} A & BR & CR & DR \\ -BR & A & D^TR & -C^TR \\ -CR & -D^TR & A & B^TR \\ -DR & C^TR & -B^TR & A \end{pmatrix}$$
where $R$ is the $n \times n$ reversal (back-identity) matrix with $R_{ij} = \delta_{i+j,\, n-1}$.
**Task:** Find four $\{+1, -1\}$ sequences of length 179 that satisfy the Goethals-Seidel condition above, yielding a Hadamard matrix of order 716.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Four sequences of length 179 with entries +1 or -1
# Each sequence is the first row of a 179x179 circulant matrix
a = [1, -1, 1, ...] # length 179
b = [1, 1, -1, ...] # length 179
c = [-1, 1, 1, ...] # length 179
d = [1, -1, -1, ...] # length 179
return {
"rows": [a, b, c, d]
} | construction | coding_theory | new_construction | 1 | null | https://oeis.org/A007299 | OEIS and recent construction databases list 716 as an unresolved Hadamard order; remains open whether H(716) exists | null | null | null | null | null |
inverse_galois_m23 | **Inverse Galois Problem for $M_{23}$**
**Definition:** The inverse Galois problem asks whether every finite group appears as the Galois group of some polynomial over $\mathbb{Q}$. The Mathieu group $M_{23}$, a sporadic simple group of order $10{,}200{,}960$, is the only sporadic simple group for which this remains unknown. All other 25 sporadic groups, including the Monster, have been realized as Galois groups over $\mathbb{Q}$.
**Task:** Construct an explicit polynomial $f(x) \in \mathbb{Z}[x]$ of degree 23 whose splitting field over $\mathbb{Q}$ has Galois group isomorphic to $M_{23}$.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Polynomial must have degree 23 (M23 acts on 23 points)
return {
"coefficients": [a0, a1, ..., a23] # for polynomial a0 + a1*x + ... + a23*x^23 in Z[x]
} | construction | number_theory | new_construction | 2 | null | https://arxiv.org/abs/2202.08222 | Häfner, 'Braid orbits and the Mathieu group M23 as Galois group' (2022). States that 'the inverse Galois problem over Q is unsolved for the Mathieu group M23' - the only sporadic simple group for which this remains open. | null | null | null | null | null |
inverse_galois_suzuki | **Inverse Galois Problem for the Suzuki Group ${}^2B_2(8)$**
**Definition:** The Suzuki group ${}^2B_2(8)$, also denoted $\mathrm{Sz}(8)$, is a simple group of order $29{,}120$. It is one of the smallest simple groups for which the inverse Galois problem over $\mathbb{Q}$ remains open. The group has a natural permutation representation of degree 65.
**Task:** Construct an explicit polynomial $f(x) \in \mathbb{Z}[x]$ whose splitting field over $\mathbb{Q}$ has Galois group isomorphic to ${}^2B_2(8)$.
**REQUIRED OUTPUT FORMAT:**
def proposed_solution():
# Polynomial must have degree 65 (Sz(8) acts on 65 points)
return {
"coefficients": [a0, a1, ..., a65] # for polynomial a0 + a1*x + ... + a65*x^65 in Z[x]
} | construction | number_theory | new_construction | 2 | null | https://pi.math.cornell.edu/~zywina/papers/smallGalois.pdf | Ranjbar and Ranjbar, 'Inverse Galois Problem For Small Simple Groups' (2025). Zywina's note explicitly lists 2B_2(8) as one of the simple groups not known to occur as a Galois group over Q\mathbb{Q}Q, and in the “open cases” summary it states it is currently unknown whether such an extension exists; it also records the order 29120. | null | null | null | null | null |
lieb_liniger_ground_state_energy_function | Consider the following open problem in mathematical physics.
**Ground-state energy function of the repulsive Lieb-Liniger Bose gas**
The (repulsive) Lieb-Liniger model describes a one-dimensional gas of bosons with contact interactions. In the thermodynamic limit, its dimensionless ground-state energy per particle can be written as a function of the (dimensionless) coupling \(\gamma>0\):
\[ \frac{E_0}{N} = \frac{\hbar^2 n_0^2}{2m}\, e(\gamma). \]
A standard characterization (see Lang et al., SciPost Phys. 3, 003 (2017), Eqs. (4)-(6)) defines \(e(\gamma)\) implicitly via an auxiliary parameter \(\alpha\ge 0\) and an unknown function \(g(z;\alpha)\) on \([-1,1]\):
1) **Lieb integral equation (Fredholm type II):**
\[
g(z;\alpha) - \frac{1}{2\pi}\int_{-1}^{1} \frac{2\alpha\, g(y;\alpha)}{\alpha^2+(y-z)^2}\,dy = \frac{1}{2\pi},\qquad z\in[-1,1].
\]
2) **Coupling relation (defines \(\alpha=\alpha(\gamma)\)):**
\[
\gamma \int_{-1}^{1} g(y;\alpha)\,dy = \alpha.
\]
3) **Energy functional:**
\[
e(\gamma) = \frac{\int_{-1}^{1} y^2\, g(y;\alpha(\gamma))\,dy}{\left(\int_{-1}^{1} g(y;\alpha(\gamma))\,dy\right)^3}.
\]
**Task:** Find a symbolic closed-form expression for the function \(e(\gamma)\) valid for \(\gamma>0\).
Despite integrability and extensive work on asymptotic/perturbative expansions, no closed-form expression for \(e(\gamma)\) as a function of \(\gamma\) is known beyond limiting regimes.
Your solution must be a finite combination of:
- Rational/algebraic numbers
- Constants: \(\pi\), \(e\), \(\gamma\) (Euler-Mascheroni), Catalan's constant, \(\zeta(n)\)
- Named special functions available in mpmath (e.g., \(\Gamma\), polylogarithms, hypergeometric functions, elliptic integrals at algebraic arguments, Bessel functions, etc.)
INADMISSIBLE (will be rejected by a compliance checker):
- Solving the integral equation numerically (direct discretization, iterative schemes, spectral methods, etc.)
- Numerical quadrature (e.g. mp.quad), numerical root-finding (e.g. mp.findroot)
- Any approximation loop whose output quality depends on iteration count (e.g. for/while refinement, nsum/nprod, truncated series tuned for accuracy)
- Returning \(e(\gamma)\) as an unevaluated integral or as an implicit definition
REQUIRED OUTPUT FORMAT:
Provide a Python function using mpmath with the exact signature:
def proposed_solution(gamma):
from mpmath import mp
mp.dps = 100
# gamma will be passed in as an mp.mpf
# Use only explicit closed-form expressions built from mp constants and named mp special functions.
result = ...
return result
| function | continuum_physics | ground_truth_computable | 1 | null | https://arxiv.org/abs/1609.08865 | Lang et al., 'Ground-state energy and excitation spectrum of the Lieb-Liniger model: accurate analytical results and conjectures about the exact solution' (SciPost Phys. 3, 003, 2017). Eqs. (4)-(6) define e(gamma) implicitly via a Fredholm integral equation and an inversion alpha(gamma). Ristivojevic (arXiv:1905.13705) emphasizes that, despite integrability, the ground-state energy function is only known in limiting cases and is otherwise accessed via high-precision numerics/expansions. Ground-truth values above were generated by a Nyström discretization of the Lieb equation with Gauss-Legendre nodes at high mp precision, then monotone inversion of gamma(alpha). | [
{
"args": [
0.5
],
"expected": "0.365979850708179424112082842345128685995014480581146505606710"
},
{
"args": [
1
],
"expected": "0.639151285272074494190862368200204800743158158448732386239999015783256975507694015"
},
{
"args": [
2
],
"expected": "1.05032... | null | null | null | null |
hensley_hausdorff_dim | Consider the following research problem at the intersection of number theory and dynamical systems.
**Hensley's Continued-Fraction Hausdorff Dimension**
**Definition:** For an integer $N \geq 2$, let $E_N \subset [0,1]$ be the set of irrationals whose continued-fraction expansion $x = [a_1, a_2, a_3, \ldots]$ satisfies $a_k \leq N$ for all $k \geq 1$. The *Hensley dimension* $d(N)$ is the Hausdorff dimension of $E_N$.
**Background:** It is classical that $d(N) \to 1$ as $N \to \infty$ (since almost all reals have unbounded partial quotients), $d(N) < 1$ for every finite $N$, and $d(N)$ is strictly increasing. However, no closed-form expression for $d(N)$ is currently known.
**Known special values:**
- $d(1) = 0$ (since $E_1$ consists of a single point, $(\sqrt{5}-1)/2$)
- $d(N) \in (0,1)$ for all $N \geq 2$
- $\lim_{N \to \infty} d(N) = 1$
**Asymptotic expansion (Hensley 1992):** As $N \to \infty$,
$$d(N) = 1 - \frac{6}{\pi^2 N} - \frac{72\ln N}{\pi^4 N^2} + O\!\left(\frac{\ln N}{N^2}\right).$$
**Task:** Propose a *closed-form* symbolic expression for $d(N)$ as a function of the positive integer $N \geq 2$.
**Inadmissible approaches (will be rejected by a compliance checker):**
- Numerical quadrature (e.g. `mp.quad`), numerical root-finding (e.g. `mp.findroot`)
- Any approximation loop whose output quality depends on iteration count (e.g. for/while refinement, `nsum`/`nprod`, truncated series tuned for accuracy)
- Matrix eigenvalue or determinant computations (e.g. transfer-operator truncation)
- Returning $d(N)$ as an unevaluated integral or as an implicit definition
**REQUIRED OUTPUT FORMAT:**
```python
def proposed_solution(N):
from mpmath import mp
mp.dps = 100
# Closed-form expression for d(N), valid for all integers N >= 2.
# No numerical root-finding, quadrature, or approximation loops.
result = ...
return result
``` | function | number_theory | ground_truth_computable | 2 | null | https://www.ams.org/journals/btran/2022-09-35/S2330-0000-2022-00109-6/S2330-0000-2022-00109-6.pdf | M. Pollicott and P. Vytnova (2022), TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 9, Pages 1102-1159. 'Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups' | [
{
"args": [
2
],
"expected": "0.53128050627720514162446"
},
{
"args": [
3
],
"expected": "0.705660908028738230607"
},
{
"args": [
4
],
"expected": "0.788945557483153972540"
},
{
"args": [
5
],
"expected": "0.836829443681208822441594... | null | null | null | null |
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