A Note on the sequence ((p-1)/2)! mod p

As a consequence of Wilson’s theorem, the factorial ((p−1)/2) ! mod p provides a square root of (−1)^((p+1)/2). The main goal of the paper is to derive an alternative expression for these square roots that avoids the factorial computation. Further, several questions concerning the infinite sequences, formed with these square roots modulo primes p congruent to 1 or congruent to 3 modulo 4, are considered and heuristics for some explanations are proposed.