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sum of powers

Source: miklos schweitzer 1995 q4

October 4, 2021
algebra

Problem Statement

For odd numbers a1,...,aka_1 , ..., a_k and even numbers b1,...,bkb_1 , ..., b_k , we know that ∑j=1kajn=∑j=1kbjn\sum_ {j = 1}^k a_j^n = \sum_{j = 1}^k b_j^n is satisfied for n = 1,2, ..., N. Prove that k≥2Nk\geq 2^N and that for k=2Nk = 2^N there exists a solution (a1,...,b1,...)(a_1,...,b_1,...) with the above properties.
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