Euler's conjecture

Euler's conjecture is a conjecture related to Fermat's Last Theorem which was proposed by Leonhard Euler in 1769. It states that for every integer n greater than 2, the sum of n-1 n-th powers of positive integers cannot itself be an n-th power.

The conjecture was disproved by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for n = 5:

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵.

In 1988, Noam Elkies found a method to construct counterexamples for the n = 4 case. His smallest counterexample was the following:

2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴.

Roger Frye subsequently found the smallest possible n = 4 counterexample by a direct computer search using techniques suggested by Elkies:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

No counterexamples for n > 5 are currently known.