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A sum which cannot be the fifth power

Source: Problem 3, Polish NO 1991

October 1, 2005
number theory unsolvednumber theory

Problem Statement

Define N=∑k=160ekkkk N=\sum\limits_{k=1}^{60}e_k k^{k^k} where ek∈{−1,1}e_k \in \{-1, 1\} for each kk. Prove that NN cannot be the fifth power of an integer.
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