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q|4^m n + 2 if pq | n^p + 2 and n + 2 | n^p + q^p

Source: Switzerland - 2017 Swiss MO Final Round p4

December 30, 2022
number theorydividesdivisible

Problem Statement

Let nn be a natural number and p,qp, q be prime numbers such that the following statements hold: pq∣np+2pq | n^p + 2 n+2∣np+qp.n + 2 | n^p + q^p. Show that there is a natural number mm such that q∣4mn+2q|4^mn + 2 holds.
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