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I have been asked to prove that $(a+b+c)\big|(a^n + b^n + c^n)$ for infinitely many $n$ given that $(a+b+c)\big| (a^2+b^2+c^2)$. Here | denotes divides (divisibility).

I have tried with the strong form of induction and tried to use the fact that $a+b+c$ also divides $2( ab+bc+ca)$, but could not prove it.

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    Have you yet considered arithmetic modulo $a+b+c$? – 2017-01-14

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