Acta Mathematica Hungarica

, Volume 142, Issue 1, pp 199–230

The Gauss–Wilson theorem for quarter-intervals


DOI: 10.1007/s10474-013-0357-1

Cite this article as:
Cosgrave, J.B. & Dilcher, K. Acta Math Hung (2014) 142: 199. doi:10.1007/s10474-013-0357-1


We define a Gauss factorial Nn! 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 

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2013

Authors and Affiliations

  1. 1.Blackrock, County DublinIreland
  2. 2.Department of Mathematics and StatisticsDalhousie UniversityHalifax, Nova ScotiaCanada