**Problem**Find six positive integers such that or prove that such integers do not exist.

Euler's sum of powers conjecture states that for the Diophantine equation does not have solutions in positive integers as soon as For it corresponds to a particular case of Fermat Last Theorem and hence is true. For and , counterexamples to the Euler's sum of powers conjecture were found by N. Elkies in 1986 and L. J. Lander, T. R. Parkin in 1966 respectively. For , no counterexamples are currently known.

* indicates original appearance(s) of problem.