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Part of 2018 Mediterranean Mathematics OIympiad

Problems(1)

Aegean integer , none of the numbers a^{n+2}+3a^n+1 with n>=1 is prime.

Source: Mediterranean 2018 P3 MMC

7/30/2018
An integer a≥1a\ge1 is called Aegean, if none of the numbers an+2+3an+1a^{n+2}+3a^n+1 with n≥1n\ge1 is prime. Prove that there are at least 500 Aegean integers in the set {1,2,…,2018}\{1,2,\ldots,2018\}.
(Proposed by Gerhard Woeginger, Austria)
number theoryprime numbersMediterranean