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.