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Extensions of the Gauss-Wilson theorem. (English) Zbl 1210.11008

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 \(M \geq 2\).

MSC:

11A07 Congruences; primitive roots; residue systems
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