Cosgrave, John B.; Dilcher, Karl Extensions of the Gauss-Wilson theorem. (English) Zbl 1210.11008 Integers 8, No. 1, Article A39, 15 p. (2008). Summary: A theorem of Gauss extending Wilson’s theorem states the congruence \((n - 1)_n ! \equiv - 1 \pmod n\) whenever \(n\) has a primitive root, and \(\equiv 1 \pmod n\) otherwise, where \(N_n!\) denotes the product of all integers up to \(N\) that are relatively prime to \(n\). In the spirit of this theorem we give a complete characterization of the multiplicative orders of \((\frac{n - 1}{2})_n! \pmod n\) for odd \(n\). In most cases we are also able to evaluate this expression explicitly \(\pmod n\), and some partial results extend to the more general case \((\frac{n - 1}{M})_n!\) for integers \(M \geq 2\). Cited in 1 ReviewCited in 5 Documents MSC: 11A07 Congruences; primitive roots; residue systems Keywords:multiplicative orders PDFBibTeX XMLCite \textit{J. B. Cosgrave} and \textit{K. Dilcher}, Integers 8, No. 1, Article A39, 15 p. (2008; Zbl 1210.11008) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n. The product of the residues in [1,n] relatively prime to n, taken modulo n. a(n) = Product_{1<=k<=n/2, gcd(k,n)=1} k. a(n) = Product_{ceiling(n/2) <= k <= n, gcd(k,n)=1} k. Phi-torial of n (A001783) modulo n. Odd numbers with at least 3 distinct prime factors. The Gauss factorial N_n! for N >= 0, n >= 1, square array read by antidiagonals.