Euler's sum of powers conjecture beyond fifth powers

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.

Problem

Task. Find an integer $k \ge 6$ and positive integers $a_1, \dots, a_{k-1}, b$ with

$$a_1^k + a_2^k + \cdots + a_{k-1}^k = b^k.$$

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.

Background

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

$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$

in 1966 [1]. For fourth powers, Elkies found counterexamples in 1988 [2], and Frye found

$$95800^4 + 217519^4 + 414560^4 = 422481^4.$$

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.

What is known

• $k = 3$: true (Fermat's Last Theorem for cubes).
• $k = 4$ and $k = 5$: false, with the counterexamples above [1, 2].
• $k \ge 6$: open. Epoch reports that experts expect the problem to be solvable, that is, a counterexample to exist.
• formal-conjectures states the open case in Lean as eulers_sum_of_powers_conjecture in EulerSumOfPowers.lean.

What would count as progress

• 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.
• 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.

References

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
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
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

Source

Imported from 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.