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:
In 1988, Noam Elkies[?] found a method to construct counterexamples for the n = 4 case. His smallest counterexample was the following:
Roger Frye[?] subsequently found the smallest possible n = 4 counterexample by a direct computer search using techniques suggested by Elkies:
No counterexamples for n > 5 are currently known.