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p=2^n+1 is prime if 3^{(p-1)/2}+1\equiv 0 \pmod p

Source: Mathcenter Contest / Oly - Thai Forum 2012 sl-2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 13, 2022
number theoryprime

Problem Statement

Let p=2n+1p=2^n+1 and 3(p−1)/2+1≡0(modp)3^{(p-1)/2}+1\equiv 0 \pmod p. Show that pp is a prime.
(Zhuge Liang)
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