Euler's sum of powers conjecture, proposed in 1769, is a generalization of Fermat's Last Theorem about the following Diophantine equation $$\sum_{i=1}^n X_i^k=Y^k\textrm{, where }n\neq 1$$
It states that for the equation to have any solutions in positive integers, $n$ must be at least $k$ (FLT is the statement that $n\ge 3$ if $k\ge 3$). For small values of $X_i,Y$, the conjecture appears to be true.
In 1966, L. J. Lander and T. R. Parkin found a counterexample for the $k=5$ case:
$$25^5+84^5+110^5+133^5=144^5.$$
In 1986, Noam Elkies found an infinite family of solutions to $X^4+Y^4+Z^4=W^4$ - another counterexample. In 1988, Roger Frye used a computer and Elkies's method to find the smallest such counterexample to the $k=4$ case:
$$95800^4+217519^4+414560^4=422481^4.$$
This is the only solution where $W,X,Y$ and $Z$ are less than $1,000,000$.