## IMC 2011 Day 1 Problem 3

Let be a prime number. Call a positive integer *interesting* if

for some polynomials .

a) Prove that the number is interesting.

b) For which is the minimal interesting number?

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Let be a prime number. Call a positive integer *interesting* if

for some polynomials .

a) Prove that the number is interesting.

b) For which is the minimal interesting number?

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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.

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