# The Gauss–Wilson theorem for quarter-intervals

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## Abstract

We define a Gauss factorial *N* _{ n }! to be the product of all positive integers up to *N* that are relatively prime to *n*. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials \(\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\) for odd positive integers *n*. The case where *n* has exactly one prime factor of the form *p*≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by *p* with the property that the order of \(\frac{p-1}{4}!\) modulo *p* is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those *n* of the above form that satisfy \(\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}\). We also report on computations that were required in the process.

### Key words and phrases

Wilson’s theorem Gauss’ theorem factorial congruence### Mathematics Subject Classification

11A07 11B65## Preview

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