Euler had conjectured that the sum of $k - 1$ many $k$-th powers is never a
$k$-th power, but this is false for $k=5$, as Lander and Parkin [LaPa67] found
$$27^5+84^5+110^5+133^5=144^5$$.
[LaPa67] Lander, L. J. and Parkin, T. R., "A counterexample to Euler's sum of powers conjecture."
Math. Comp. (1967), 101--103.
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