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Find all pairs of prime numbers (p, q)

Source: KJMO 2019 p3

January 8, 2021
number theoryprime numbersKJMO

Problem Statement

Find all pairs of prime numbers p, q(p≤q)p,\,q(p\le q) satisfying the following condition: There exists a natural number nn such that 2n+3n+⋯+(2pq−1)n2^{n}+3^{n}+\cdots+(2pq-1)^{n} is a multiple of 2pq2pq.
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