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m \ne k_1^n+ k_2^n+... k_n^n, m not equal to sum of perfect powers

Source: I Soros Olympiad 1994-95 Ukraine R2 11.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 6, 2024
number theoryPerfect Powers

Problem Statement

Prove that for any natural n>1n>1 there are infinitely many natural numbers mm such that for any nonnegative integers k1k_1,k2k_2, ......,kmk_m, m≠k1n+k2n+...knn,m \ne k_1^n+ k_2^n+... k_n^n,
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