Problem
NT-B2-M04-P018 Product of Inverse Pairs
#18
★★★★★ Level 5 of 5
Let \(p>3\) be prime. Find the product of all \(x\in\{1,\ldots,p-1\}\) such that \(x\not\equiv x^{-1}\pmod p\), modulo \(p\).
Remove the self-inverse elements \(1\) and \(-1\).
The self-inverse elements satisfy \(x^2\equiv1\pmod p\), so they are \(1\) and \(-1\). The product of all nonzero residues is \((p-1)!\equiv-1\pmod p\). Removing the factors \(1\) and \(-1\), the remaining product is \((-1)/(1\cdot(-1))\equiv1\pmod p\).
Checks understanding of Wilson's proof.