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p divides a^2+ab+b^2 and a^n+b^n+c^n, but not a+b+c, prove n, p-1 not coprime

Source: Danube 2013 p2

July 22, 2019
number theorycoprimeprimedivides

Problem Statement

Let a,b,c,na, b, c, n be four integers, where n≥2\ge 2, and let pp be a prime dividing both a2+ab+b2a^2+ab+b^2 and an+bn+cna^n+b^n+c^n, but not a+b+ca+b+c. for instance, a≡b≡−1(mod  3),c≡1(mod  3),na \equiv b \equiv -1 (mod \,\, 3), c \equiv 1 (mod \,\, 3), n a positive even integer, and p=3p = 3 or a=4,b=7,c=−13,n=5a = 4, b = 7, c = -13, n = 5, and p=31p = 31 satisfy these conditions. Show that nn and p−1p - 1 are not coprime.
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