Prove that for each prime , there exists a positive integer n and integers , not divisible by p such that
We claim that satisfies the conditions of the problem. We first consider a system of equations:
We repeatedly use the most well-known Pythagorean triple to obtain the following equalities:
Indeed, we set
, for every , and .
To finish our proof, we only need to note that by Fermat’s Little Theorem, we have
note that there are infinitely many such n, for instance all multiples of .
Ref: 104 Problems Number Theory Problems (from the training of the USA IMO Team) by Prof. Titu Andreescu, Dorin Andrica and Zumin Feng.