Sum of Three Cubes
Sum of Three Cubes. Moderately interesting problem from Epoch AI’s FrontierMath: Open Problems; answers are checked by a verifier program.
Epoch field label “Diophantine” mapped to MSC2020 11D25 for Sum of Three Cubes. https://epoch.ai/frontiermath/open-problems/sum-of-three-cubes Not an author-supplied code.
Problem statement
The sum of three cubes problem asks which positive integers can be expressed as the sum of three cubes. In other words, for which positive integers do there exist integers such that ? Three is such an example:
and
It is known that if is congruent to 4 or 5 mod 9, then cannot be written as the sum of three cubes. But as far as we know, this could be the only obstruction: possibly every number not congruent to 4 or 5 mod 9 can be written as the sum of three cubes.
See this Numberphile video for more about writing 3 as the sum of three cubes, and this Numberphile video for more about writing 42 as the sum of three cubes; the solution for 42 was found only in 2019. Many of the new results on the sum of cubes problem, including the solution for 42, were found using computational methods.
The smallest number for which it is unknown whether can be written as the sum of three cubes is . This problem asks for three integers such that . We also ask for a new (i.e. previously unknown) way of writing 3 as the sum of three cubes.
Since lots of computational search has gone into the sum of three cubes problem, it is possible that a clever idea is needed for AI to solve this problem.
Read more about the 114 case here.
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
FrontierMath: Open Problems lists this problem with notability “Moderately interesting”, field “Diophantine” and type “Integer Solution; Record Breaking”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.
Verifiable task (n = 3) — Epoch prompt, verbatim
The "sum of three cubes" problem asks which integers n can be written as
x^3 + y^3 + z^3 = n
where x, y, z are integers (positive, negative, or zero).
Three representations of n = 3 are already known: (1, 1, 1), (4, 4, -5), and the triple found by Booker and Sutherland in 2019. Your task is to find a *new* one: any triple summing to 3 that is not one of these three (in any order) will be accepted. Whether a fourth representation exists is an open problem.
## Task
Find integers x, y, z such that
x^3 + y^3 + z^3 = 3
## Submission
Submit your three integers as (x, y, z) using the submit tool. There is no bound on the magnitude of x, y, or z. Your submission is checked exactly with arbitrary-precision arithmetic, so an approximate or rounded answer will not be accepted.
Verifiable task (n = 114) — Epoch prompt, verbatim
The "sum of three cubes" problem asks which integers n can be written as
x^3 + y^3 + z^3 = n
where x, y, z are integers (positive, negative, or zero).
## Task
Find integers x, y, z such that
x^3 + y^3 + z^3 = 114
## Submission
Submit your three integers as (x, y, z) using the submit tool. There is no bound on the magnitude of x, y, or z. Your submission is checked exactly with arbitrary-precision arithmetic, so an approximate or rounded answer will not be accepted.
Progress & source assessment
Epoch lists this problem as unsolved in its data archive updated 7 October 2026. Earlier AI attempts are not displayed by Epoch.
- [1]Epoch AI, ‘FrontierMath: Open Problems’. Published online at epoch.ai. Retrieved from https://epoch.ai/frontiermath/open-problems (data updated 7 October 2026). — Sum of Three Cubes linkCC BY 4.0 (Epoch AI data and problem pages); attributed reproduction of problem descriptions and verbatim task prompts.
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
Machine attempts
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Discussion
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