OC634 from Crux Mathematicorum Vol. 49, No. 5

OC634 (from Crux Mathematicorum Vol. 49, No. 5)

Prove that for infinitely many integers \(n > 1\) the equation \[(x + 1)^{n+1} - (x - 1)^{n+1} = y^n\] has no integer solutions.

Proof

Let $p = n + 1 $. By the binomial expansion, we rewrite the LHS of equation as

\[(x + 1)^{p} - (x - 1)^{p} = \sum_{k=0}^{p} \binom{p}{k} x^{p-k} - \sum_{k=0}^{p} \binom{p}{k} (-1)^k x^{p-k} = 2 \sum_{\text{ k is odd}} \binom{p}{k} x^{p-k}. \]

Note that when \(p\) is an odd prime number, \[\sum_{\text{ k is odd}} \binom{p}{k} x^{p-k} = 1 + \sum_{k=1,3,\ldots,p-2} \binom{p}{k} x^{p-k}. \] Noticing that $ p | $ for \(1 < k < p\) when \(p\) is a prime, we claim that \[(x + 1)^{p} - (x - 1)^{p} \equiv 2 \pmod {p}\] if \(p\) is an odd prime number.

However, we either have \(y^{p-1} \equiv 0 \pmod p\) if \(y\) is a multiple of \(p\), or \(y^{p-1} \equiv 1 \pmod p\) if \(y\) is coprime to \(p\) by Fermat’s little theorem. Hence the proposed equation has no integer solution if \(p = n+1\) is an odd prime number.

Since there are infinitely many odd prime numbers, we proved that for infinitely many integers \(n > 1\) the equation \[(x + 1)^{n+1} - (x - 1)^{n+1} = y^n\] has no integer solutions.