Eulers conjecture

From Exampleproblems

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 nth powers of positive integers cannot itself be an nth power.

The conjecture was disproven 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.

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