Home > Olympiad > IMC 2011 Day 1 Problem 3

IMC 2011 Day 1 Problem 3


Let p be a prime number. Call a positive integer n  interesting if

x^n-1=(x^p-x+1)f(x)+pg(x) for some polynomials f,g \in \Bbb{Z}[X].

a) Prove that the number p^p-1 is interesting.

b) For which p is p^p-1 the minimal interesting number?

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  1. Vijay Joshi
    August 2, 2011 at 7:04 am

    b) When p = 2, p^p – 1 is 3, consider x^3 – 1 which can be written as (x^2 – x +1)(x+1) + 2 ( 0) , therefore p=2 is the number when p^p – 1 is minimal interesting number.

    • August 2, 2011 at 9:47 am

      Again, it is not a complete answer. Indeed, for p=2 we have that p^p-1 is the minimal interesting number. The idea is to find all prime numbers p such that p^p-1 is the minimal interesting number when p is fixed.

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