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.