(Turkey 1997)

Prove that for each prime , there exists a positive integer n and integers , not divisible by p such that

Proof:

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.

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Nalin Pithwa