dataset_index int64 | paper_id string | question_link string | original_question string | self_contained_problem string | taxonomy_level_1 string | taxonomy_level_2 string | taxonomy_level_3 string | open_status string | status_search_result string | status_evidence string | status_evidence_urls string | generation list | finish_reason list | stop_reason list | generated_token_count list | reasoning_token_count list |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 07-workshop-problems | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf#page=1 | Determine all integer pairs (x, y) that satisfy the Diophantine equation x^2 - x = y^5 - y. | "Determine all pairs of integers $(x, y) \\in \\mathbb{Z} \\times \\mathbb{Z}$ that satisfy the Diop(...TRUNCATED) | Number Theory | Number Theory | Diophantine equations | unknown | "The equation appears as the first open problem in a 2007 Leiden/Lorentz Center workshop problem lis(...TRUNCATED) | "[{\"title\": \"Some open problems about Diophantine equations\", \"url\": \"https://pub.math.leiden(...TRUNCATED) | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf | ["We need to solve the Diophantine equation x^2 - x = y^5 - y over integers. So we want all integer (...TRUNCATED) | [
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"stop",
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null
] | [
57622,
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1 | 0704.1689 | https://arxiv.org/pdf/0704.1689.pdf#page=2 | "Let P(z) be a homogeneous polynomial of degree 4 in C[z] whose Hessian matrix Hes P is nilpotent, s(...TRUNCATED) | "Let n be a positive integer and let C[z] = C[z_1, ..., z_n] denote the polynomial ring in n variabl(...TRUNCATED) | Algebra and Representation Theory | Algebra | Hessian nilpotent polynomials | open | "The statement is the degree-4 Vanishing Conjecture introduced by Wenhua Zhao, shown to be equivalen(...TRUNCATED) | "[{\"title\": \"Hessian Nilpotent Polynomials and the Jacobian Conjecture\", \"url\": \"https://arxi(...TRUNCATED) | https://arxiv.org/abs/math/0409534 https://www.impan.pl/en/publishing-house/journals-and-series/annales-polonici-mathematici/all/93/2/85371/some-properties-of-and-open-problems-on-hessian-nilpotent-polynomials https://arxiv.org/pdf/1308.0674 | ["We need to analyze the problem.\n\nLet n ≥ 1 (positive integer). C[z] = C[z_1, ..., z_n]. For P (...TRUNCATED) | [
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] | [
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] | [
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2 | 07-workshop-problems | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf#page=1 | "Determine all integer pairs (x, y) for which the binomial coefficients satisfy C(x, 2) = C(y, 5), e(...TRUNCATED) | "Determine all pairs of integers $(x,y)$ satisfying the equation\n$$\\binom{x}{2}=\\binom{y}{5},$$\n(...TRUNCATED) | Number Theory | Number Theory | Diophantine equations with binomial coefficients | solved | "The equation $\\binom{x}{2}=\\binom{y}{5}$ defines a curve of genus 2, which falls outside the elli(...TRUNCATED) | "[{\"title\": \"Elliptic binomial diophantine equations (Stroeker–de Weger)\", \"url\": \"https://(...TRUNCATED) | https://www.ams.org/journals/mcom/1999-68-227/S0025-5718-99-01047-9/ https://arxiv.org/abs/1901.03841 | ["We need to solve the equation:\n\n\\[\\binom{x}{2} = \\binom{y}{5},\\]\n\nwhere the binomial coeff(...TRUNCATED) | [
"stop",
"stop",
"stop",
"stop"
] | [
null,
null,
null,
null
] | [
48419,
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] |
3 | 0704.1689 | https://arxiv.org/pdf/0704.1689.pdf#page=15 | "Suppose P(z) is a formal power series in C[[z]] with order at least 2 and with (Hes P)(0) nilpotent(...TRUNCATED) | "Work over the formal power series ring $\\mathbb{C}[[z]] = \\mathbb{C}[[z_1,\\ldots,z_n]]$. For $P((...TRUNCATED) | Algebra and Representation Theory | Algebra | Hessian nilpotent polynomials | unknown | "A web search did not surface a published resolution of this specific question (whether all self-inv(...TRUNCATED) | "[{\"title\": \"Some properties of and open problems on Hessian nilpotent polynomials\", \"url\": \"(...TRUNCATED) | https://www.impan.pl/en/publishing-house/journals-and-series/annales-polonici-mathematici/all/93/2/85371/some-properties-of-and-open-problems-on-hessian-nilpotent-polynomials https://arxiv.org/abs/math/0409534 https://www.sciencedirect.com/science/article/pii/S0022404908000480 | ["We need to solve the problem: Over formal power series ring C[[z]] with z=(z1,...,zn). For P(z) wi(...TRUNCATED) | [
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"stop",
"stop"
] | [
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] | [
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4 | 07-workshop-problems | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf#page=1 | "Extend Ellenberg's approach so as to solve the Diophantine equation x^2 + y^6 = z^n for integers x,(...TRUNCATED) | "For every integer $n \\geq 3$, determine all triples of integers $(x, y, z)$ with $\\gcd(x,y,z)=1$ (...TRUNCATED) | Number Theory | Number Theory | generalized Fermat equations | solved | "The equation $x^2+y^6=z^n$ was resolved by M. A. Bennett and I. Chen in \"Multi-Frey Q-curves and t(...TRUNCATED) | "[{\"title\": \"Multi-Frey Q-curves and the Diophantine equation a^2+b^6=c^n\", \"url\": \"https://p(...TRUNCATED) | https://personal.math.ubc.ca/~bennett/BeCh.pdf https://personal.math.ubc.ca/~bennett/publ.html | ["We need to classify all integer solutions (x,y,z) with gcd(x,y,z)=1 and xyz ≠ 0 satisfying x^2 +(...TRUNCATED) | [
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"stop",
"stop",
"stop"
] | [
null,
null,
null,
null
] | [
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5 | 07-workshop-problems | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf#page=1 | "Determine whether one can solve the Diophantine equation x^2 - 2 = y^p for integers x and y and for(...TRUNCATED) | "Determine all integer solutions $(x, y, p)$ with $p$ a prime number satisfying $p \\geq 3$ to the e(...TRUNCATED) | Number Theory | Number Theory | Lebesgue–Nagell equations | partially_solved | "The equation x^2 - 2 = y^p has received substantial attention. It has been resolved for small prime(...TRUNCATED) | "[{\"title\": \"On the Lebesgue-Nagell equation x^2-2 = y^p\", \"url\": \"https://arxiv.org/abs/2507(...TRUNCATED) | https://arxiv.org/abs/2507.12397 https://arxiv.org/abs/math/0405220 | ["We need to solve $x^2 - 2 = y^p$ for integer $x, y$, prime $p \\ge 3$. Find all integer solutions.(...TRUNCATED) | [
"stop",
"stop",
"stop",
"stop"
] | [
null,
null,
null,
null
] | [
74111,
57710,
73975,
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] | [
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] |
6 | 0704.1689 | https://arxiv.org/pdf/0704.1689.pdf#page=19 | "Does there exist a Hessian nilpotent formal power series P(z) in C[[z]] that is not a polynomial an(...TRUNCATED) | "Let $\\mathbb{C}[[z]] = \\mathbb{C}[[z_1,\\dots,z_n]]$ denote the ring of formal power series in $n(...TRUNCATED) | Algebra and Representation Theory | Algebra | Hessian nilpotent power series and the Vanishing Conjecture | open | "No counterexample of a non-polynomial Hessian nilpotent formal power series violating the Vanishing(...TRUNCATED) | "[{\"title\": \"Some properties of and open problems on Hessian nilpotent polynomials\", \"url\": \"(...TRUNCATED) | https://www.impan.pl/en/publishing-house/journals-and-series/annales-polonici-mathematici/all/93/2/85371/some-properties-of-and-open-problems-on-hessian-nilpotent-polynomials https://ar5iv.labs.arxiv.org/html/0704.1691 | ["We need to analyze the problem carefully. \n\nWe have the ring of formal power series in n variabl(...TRUNCATED) | [
"stop",
"stop",
"stop",
"stop"
] | [
null,
null,
null,
null
] | [
87135,
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7 | 07-workshop-problems | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf#page=1 | "Do there exist prime numbers q such that, for every sufficiently large prime exponent p, the equati(...TRUNCATED) | "Determine whether there exists a prime number $q$ with the following property: for every sufficient(...TRUNCATED) | Number Theory | Number Theory | Generalized Fermat equations | unknown | "Searches on the generalized Fermat equation of signature (2,3,p) indicate active progress on cases (...TRUNCATED) | "[{\"title\": \"Generalised Fermat equation: a survey of solved cases\", \"url\": \"https://arxiv.or(...TRUNCATED) | https://arxiv.org/pdf/2412.11933 https://arxiv.org/abs/1703.05058 | ["We are asked: \"Determine whether there exists a prime number q with the following property: for e(...TRUNCATED) | [
"stop",
"stop",
"stop",
"stop"
] | [
null,
null,
null,
null
] | [
8182,
13695,
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] | [
8030,
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] |
8 | 0704.1689 | https://arxiv.org/pdf/0704.1689.pdf#page=2 | "Let P(z) be a formal power series in C[[z]] with order o(P(z)) >= 2. If the sequence Delta^m(P(z)^m(...TRUNCATED) | "Let $P(z) = P(z_1, \\ldots, z_n) \\in \\mathbb{C}[[z_1, \\ldots, z_n]]$ be a formal power series in(...TRUNCATED) | Algebra and Representation Theory | Algebra | Hessian nilpotent polynomials | open | "This is a conjecture posed by Wenhua Zhao in the context of the Vanishing Conjecture, which is know(...TRUNCATED) | "[{\"title\": \"Some properties of and open problems on Hessian nilpotent polynomials\", \"url\": \"(...TRUNCATED) | https://www.impan.pl/en/publishing-house/journals-and-series/annales-polonici-mathematici/all/93/2/85371/some-properties-of-and-open-problems-on-hessian-nilpotent-polynomials https://arxiv.org/abs/math/0409534 | ["We need to analyze the problem:\n\nLet P(z) be a formal power series in n variables over C, order (...TRUNCATED) | [
"stop",
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"stop"
] | [
null,
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null,
null
] | [
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] | [
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9 | 07-workshop-problems | https://pub.math.leidenuniv.nl/~evertsejh/07-workshop-problems.pdf#page=1 | Determine whether Kraus' equation x^3 + y^3 = z^p can be solved for all prime exponents p >= 3. | "Let p be a prime number with p ≥ 3. Consider the Diophantine equation\n x^3 + y^3 = z^p\nto be (...TRUNCATED) | Number Theory | Number Theory | Generalized Fermat equations | partially_solved | "The equation x^3 + y^3 = z^p has been resolved for many primes p via modular methods (e.g., Kraus, (...TRUNCATED) | "[{\"title\": \"On the Fermat-type Equation x^3 + y^3 = z^p\", \"url\": \"https://arxiv.org/abs/1601(...TRUNCATED) | https://arxiv.org/abs/1601.06361 https://arxiv.org/pdf/2412.11933 | ["We need to solve the Diophantine equation x^3 + y^3 = z^p for primes p ≥ 3, with integers x, y, (...TRUNCATED) | [
"stop",
"stop",
"stop",
"stop"
] | [
null,
null,
null,
null
] | [
32715,
34845,
33852,
32779
] | [
30923,
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32216,
31241
] |
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