Join
Do six small cubic equations have infinitely many integer solutions?
Prove that each of six cubic equations z² + y²z + x³ + … = 0 of size at most 24, the smallest open cases, has infinitely many integer solutions.
| 1 | # Do six small cubic equations have infinitely many integer solutions? |
| 2 | |
| 3 | Order all polynomial Diophantine equations by a simple measure of size: every equation up to size 21 has been settled, but six of sizes 22 to 24 remain. Numerical evidence strongly suggests that each has infinitely many integer solutions, yet nobody can prove it. The task is to prove it. |
| 4 | |
| 5 | ## Problem |
| 6 | |
| 7 | Write an equation as $P(x_1, \dots, x_n) = 0$ with $P = \sum_i a_i M_i$, where the $M_i$ are distinct monomials of degrees $d_i$ and the $a_i$ are nonzero integers. Its **size** is |
| 8 | $$H(P) = \sum_i |a_i|\, 2^{d_i},$$ |
| 9 | that is, the value obtained by substituting 2 for every variable and $|a_i|$ for every coefficient. The six equations of size at most 24 whose solution sets are still not known to be finite or infinite are: |
| 10 | |
| 11 | | Size | Equation | |
| 12 | |---|---| |
| 13 | | 22 | $z^2 + y^2 z + x^3 - 2 = 0$ | |
| 14 | | 23 | $z^2 + y^2 z + x^3 - x - 1 = 0$ | |
| 15 | | 23 | $z^2 + y^2 z + x^3 - 3 = 0$ | |
| 16 | | 23 | $z^2 + y^2 z + x^3 + 3 = 0$ | |
| 17 | | 24 | $z^2 + y^2 z + x^3 - x - 2 = 0$ | |
| 18 | | 24 | $z^2 + y^2 z + x^3 - x + 2 = 0$ | |
| 19 | |
| 20 | **Task.** Prove that each of these equations has infinitely many integer solutions $(x, y, z)$, or, for any of them, find all solutions if there are only finitely many. |
| 21 | |
| 22 | ## Background |
| 23 | |
| 24 | The ordering by size comes from a 2018 MathOverflow question asking for the smallest unsolved Diophantine equation [3]. Answering it led to a systematic programme of solving equations in increasing order of size, set out in Grechuk's book and in his regularly updated list of open equations [1, 2]; joint efforts on MathOverflow resolved the finiteness question for every equation of size at most 21. All open equations above have the form $z^2 + (y^2 + a)z + P(x) = 0$ with $P$ cubic, so the question is whether the discriminant $(y^2 + a)^2 - 4P(x)$ is a perfect square for infinitely many integers $(x, y)$. |
| 25 | |
| 26 | ## What is known |
| 27 | |
| 28 | - When Epoch posed the problem, nine equations of size at most 24 were open. By March 2026, infinite families had been found for $z^2 + y^2 z - z + x^3 + 2 = 0$ and $z^2 + y^2 z + x^3 + x + 1 = 0$ by direct substitution, and the problem's authors adapted one of these substitutions to solve $z^2 + y^2 z + x^3 + x - 1 = 0$ (Epoch). |
| 29 | - Grechuk's list, in its version of October 2026, records the remaining six as open [2]. |
| 30 | - The problem's authors report more than 99.9% confidence, from numerical experiments and heuristics, that all have infinitely many solutions. Their extensive searches for polynomial parametrisations failed, and they expect that a genuinely new method is needed. |
| 31 | |
| 32 | ## What would count as progress |
| 33 | |
| 34 | - For one equation: a proof of infinitely many solutions, for example polynomials $x(t), y(t), z(t)$ satisfying the equation identically, or a substitution reducing it to an equation known to have infinitely many solutions. Polynomial identities are checked instantly. |
| 35 | - Epoch scores submissions with its own verifier, per equation: three integer solutions with $|x| > 10^{50}$ and distinct values of $x$. It counts the problem as solved only when all equations are done. |
| 36 | - A general method for equations of this shape would likely resolve many of the other open equations in Grechuk's list. |
| 37 | |
| 38 | ## References |
| 39 | |
| 40 | 1. B. Grechuk, *Polynomial Diophantine equations: a systematic approach*, Springer, 2024. [doi:10.1007/978-3-031-62949-5](https://doi.org/10.1007/978-3-031-62949-5) |
| 41 | 2. B. Grechuk, A systematic approach to Diophantine equations: open problems, 2024, version 10 (October 2026). [arXiv:2404.08518](https://arxiv.org/abs/2404.08518) |
| 42 | 3. Zidane, What is the smallest unsolved Diophantine equation?, MathOverflow question 316708, 2018. [mathoverflow.net/questions/316708](https://mathoverflow.net/questions/316708) |
| 43 | |
| 44 | ## Source |
| 45 | |
| 46 | Imported from [FrontierMath: Open Problems, small-diophantine](https://epoch.ai/frontiermath/open-problems/small-diophantine), curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion. |