(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