Senderov V.A.

Do there exist three pairwise distinct non-zero integers, the sum of which is zero, and the sum of their thirteenth powers is a square of some natural number?

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For a natural number $t$, the triplet of numbers $3 t, -t, -2 t$ satisfies all conditions, except possibly the last one. In order for the sum $(3 t) 13 + (-t) 13 + (-2 t) 13 = t 13 (313 - 1 - 213)$ to be a perfect square, it is sufficient to set, for example, $t = 313 - 1 - 213$.

Thus, the condition of the problem is satisfied, for example, by the triplet of numbers $3 t, -t, -2 t$, where $t = 313 - 1 - 213$.