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Roots raised to the p-th power

Source: Kvant Magazine No. 10 2021 M2671

March 9, 2023
algebranumber theoryKvant

Problem Statement

Let x1x_1 and x2x_2 be the roots of the equation x2−px+1=0x^2-px+1=0 where p>2p>2 is a prime number. Prove that x1p+x2px_1^p+x_2^p is an integer divisible by p2p^2.
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