# 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:

- 27
^{5} + 84^{5} + 110^{5} + 133^{5} = 144^{5}.

In

1988, Noam Elkies found a method to construct counterexamples for the

*n* = 4 case. His smallest counterexample was the following:

- 2682440
^{4} + 15365639^{4} + 18796760^{4} = 20615673^{4}.

Roger Frye subsequently found the smallest possible

*n* = 4 counterexample by a direct computer search using techniques suggested by Elkies:

- 95800
^{4} + 217519^{4} + 414560^{4} = 422481^{4}.

No counterexamples for

*n* > 5 are currently known.