Euler’s conjecture

In 1769, Leonhard Euler conjectured that for there is no set of positive integers $a_1,\mathrm{\dots },a_k$ such that

$$\sum _{i=1}^{k}a_i{}^n=m^n,$$

where $m$ is an integer. Lander and Parkin in 1966 disproved the conjecture with this $k=4,n=5$ counterexample: ${27}^5+{84}^5+{110}^5+{133}^5={144}^5$. More counterexamples have been discovered since then.

Title	Euler’s conjecture
Canonical name	EulersConjecture
Date of creation	2013-03-22 16:25:43
Last modified on	2013-03-22 16:25:43
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	7
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 11B13
Classification	msc 11D41
Synonym	Euler conjecture