2013-08 Minimum of a set involving polynomials with integer coefficients

Let \( p \) be a prime number. Let \( S_p \) be the set of all positive integers \( n \) satisfying
\[
x^n – 1 = (x^p – x + 1) f(x) + p g(x)
\]
for some polynomials \( f \) and \( g \) with integer coefficients. Find all \( p \) for which \( p^p -1 \) is the minimum of \( S_p \).

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2013-08 Minimum of a set involving polynomials with integer coefficients, 4.9 out of 5 based on 10 ratings