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x_1^n + x_2^n and x_1^{n+1} + x_2^{n+1} are integer and coprime.

Source: Polish MO Finals 1965 p2

August 30, 2024
algebranumber theorycoprimeInteger

Problem Statement

Prove that if the numbers x1 x_1 and x2 x_2 are roots of the equation x2+px−1=0 x^2 + px - 1 = 0 , where p p is an odd number, then for every natural nnnumber x1n+x2n x_1^n + x_2^n and x1n+1+x2n+1 x_1^{n+1} + x_2^{n+1} are integer and coprime.
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