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IMC 2011 Day 1 Problem 3

Source:

July 30, 2011
algebrapolynomialsuperior algebrasuperior algebra unsolved

Problem Statement

Let pp be a prime number. Call a positive integer nn interesting if xn−1=(xp−x+1)f(x)+pg(x)x^n-1=(x^p-x+1)f(x)+pg(x) for some polynomials ff and gg with integer coefficients. a) Prove that the number pp−1p^p-1 is interesting. b) For which pp is pp−1p^p-1 the minimal interesting number?
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