Euler's sum of powers conjecture beyond fifth powers

Find k−1 positive k-th powers summing to a k-th power for some k ≥ 6, extending the counterexamples to Euler's conjecture for k = 4 and 5.

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1	# Euler's sum of powers conjecture beyond fifth powers
2
3	In 1769 Euler conjectured that a $k$-th power can never be written as a sum of fewer than $k$ positive $k$-th powers. A computer search refuted it for fifth powers in 1966, and counterexamples for fourth powers followed in 1988, but for every exponent $k \ge 6$ no counterexample is known. The task is to find one.
4
5	## Problem
6
7	Task. Find an integer $k \ge 6$ and positive integers $a_1, \dots, a_{k-1}, b$ with
8	$$a_1^k + a_2^k + \cdots + a_{k-1}^k = b^k.$$
9
10	Euler's conjecture forbids any sum of fewer than $k$ positive $k$-th powers that is itself a $k$-th power. Exactly $k - 1$ terms is the boundary case; a solution with fewer terms would refute it as well.
11
12	## Background
13
14	For $k = 3$ the statement is a case of Fermat's Last Theorem and is true. Euler's conjecture stood for almost two centuries until Lander and Parkin's computer search found
15	$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$
16	in 1966 [1]. For fourth powers, Elkies found counterexamples in 1988 [2], and Frye found
17	$$95800^4 + 217519^4 + 414560^4 = 422481^4.$$
18	Lander, Parkin and Selfridge surveyed the wider subject of equal sums of like powers [3]. For $k \ge 6$ nothing is known either way: no counterexample, and no proof that none exists.
19
20	## What is known
21
22	- $k = 3$: true (Fermat's Last Theorem for cubes).
23	- $k = 4$ and $k = 5$: false, with the counterexamples above [1, 2].
24	- $k \ge 6$: open. Epoch reports that experts expect the problem to be solvable, that is, a counterexample to exist.
25	- formal-conjectures states the open case in Lean as `eulers_sum_of_powers_conjecture` in [EulerSumOfPowers.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Wikipedia/EulerSumOfPowers.lean).
26
27	## What would count as progress
28
29	- A list $(a_1, \dots, a_{k-1}, b)$ of positive integers; as in Epoch's verifier, $k$ is read from its length and the identity is checked exactly with big-integer arithmetic, in microseconds.
30	- A parametric family, or a method that generates many solutions for some $k \ge 6$, would be worth far more than a single sporadic solution found by brute force.
31
32	## References
33
34	1. L. J. Lander and T. R. Parkin, Counterexample to Euler's conjecture on sums of like powers, Bull. Amer. Math. Soc. 72 (1966), 1079. [doi:10.1090/S0002-9904-1966-11654-3](https://doi.org/10.1090/S0002-9904-1966-11654-3)
35	2. N. D. Elkies, On $A^4 + B^4 + C^4 = D^4$, Math. Comp. 51 (1988), 825–835. [doi:10.1090/S0025-5718-1988-0930224-9](https://doi.org/10.1090/S0025-5718-1988-0930224-9)
36	3. L. J. Lander, T. R. Parkin and J. L. Selfridge, A survey of equal sums of like powers, Math. Comp. 21 (1967), 446–459. [doi:10.1090/S0025-5718-1967-0222008-0](https://doi.org/10.1090/S0025-5718-1967-0222008-0)
37
38	## Source
39
40	Imported from [FrontierMath: Open Problems, euler-sum-of-powers](https://epoch.ai/frontiermath/open-problems/euler-sum-of-powers), 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.