Euler's Sum of Powers Conjecture
Euler's Sum of Powers Conjecture. Moderately interesting problem from Epoch AI’s FrontierMath: Open Problems; answers are checked by a verifier program.
Epoch field label “Diophantine” mapped to MSC2020 11D41 for Euler's Sum of Powers Conjecture. https://epoch.ai/frontiermath/open-problems/euler-sum-of-powers Not an author-supplied code.
Problem statement
This problem asks for a counterexample to Euler’s sum of powers conjecture. Specifically, the problem asks: for some , find nonzero integers such that .
Euler conjectured that no solution to the above equation exists for . The conjecture is true for (which is a special case of Fermat’s Last Theorem), but false for and . No counterexample for is known.
Experts expect that this problem is solvable, and the verifier can check a correct counterexample. The main risk with this problem is a brute-force solution for a single value of may not be very interesting or insightful. An ideal solution would either be a construction that works for all , or an efficient way of generating counterexamples.
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 “Counterexample; Integer Solution”. Each problem has a bespoke verifier program, so a proposed answer can be checked automatically.
Verifiable task — Epoch prompt, verbatim
In 1769 Leonhard Euler conjectured that for every integer n > 2, no sum of fewer than n positive n-th powers can itself be an n-th power. In its tightest form this says the equation
a_1^n + a_2^n + ... + a_{n-1}^n = b^n
has no solution in positive integers: with only n-1 terms on the left, an n-th power cannot be reached. Euler's conjecture stood for nearly two centuries. It was finally refuted for n = 5 by Lander and Parkin in 1966, who found
27^5 + 84^5 + 110^5 + 133^5 = 144^5,
and later for n = 4 by Elkies and Frye. But for every exponent n > 5, no counterexample is known, and whether any exists remains open.
## Task
Find an integer k > 5 and positive integers a_1, ..., a_{k-1}, a_k such that
a_1^k + a_2^k + ... + a_{k-1}^k = a_k^k.
The length of your list determines the exponent: if you submit k integers, the exponent is k, the first k-1 are summed, and the last is the right-hand side. So a valid submission of k integers is exactly a counterexample to Euler's conjecture for exponent k.
## Submission
Submit your list of integers (a_1, ..., a_k) using the submit tool. Every integer must be positive, as in the statement of the conjecture; zero or negative entries are rejected. There is no bound on their magnitude. Your submission is checked exactly with arbitrary-precision integer 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). — Euler's Sum of Powers Conjecture 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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