Acta Mathematica Hungarica

, Volume 142, Issue 1, pp 199–230

# The Gauss–Wilson theorem for quarter-intervals

• John B. Cosgrave
• Karl Dilcher
Article

## 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

11A07 11B65

## References

1. [1]
B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, Wiley (New York, 1998).
2. [2]
J. B. Cosgrave and K. Dilcher, Extensions of the Gauss–Wilson theorem, Integers, 8 (2008), A39, available at http://www.integers-ejcnt.org/vol8.html.
3. [3]
J. B. Cosgrave and K. Dilcher, Mod p 3 analogues of theorems of Gauss and Jacobi on binomial coefficients, Acta Arith., 142 (2010), 103–118.
4. [4]
J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, Int. J. Number Theory, 7 (2011), 145–171.
5. [5]
J. B. Cosgrave and K. Dilcher, An introduction to Gauss factorials, Amer. Math. Monthly, 118 (2011), 810–828.
6. [6]
J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, II, in preparation. Google Scholar
7. [7]
R. E. Crandall, A general purpose factoring program. Perfectly Scientific – the algorithm company, available from http://www.perfsci.com/free-software.asp.
8. [8]
L. E. Dickson, History of the Theory of Numbers. Volume I: Divisibility and Primality, Chelsea (New York, 1966). Google Scholar
9. [9]
P. Gaudry, A. Kruppa, F. Morain, L. Muller, E. Thomé and P. Zimmermann, cado-nfs, An Implementation of the Number Field Sieve Algorithm. Release 1.0, available from http://cado-nfs.gforge.inria.fr/.
10. [10]
H. W. Gould, Combinatorial Identities, revised ed., Gould Publications (Morgantown, W.Va, 1972).
11. [11]
D. H. Lehmer, The distribution of totatives, Canad. J. Math., 7 (1955), 347–357.
12. [12]
L. J. Mordell, The congruence (p−1/2)!≡±1(modp), Amer. Math. Monthly, 68 (1961), 145–146.
13. [13]
F. Morain, Private communication (2006). Google Scholar
14. [14]
I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th ed., Wiley (1991). Google Scholar
15. [15]
The On-Line Encyclopedia of Integer Sequences, http://oeis.org/.
16. [16]
P. Ribenboim, The Little Book of Bigger Primes, 2nd ed., Springer-Verlag (New York, 2004).
17. [17]
E. W. Weisstein, Lucas Sequence. From MathWorld – A Wolfram Web Resource, http://mathworld.wolfram.com/LucasSequence.html.
18. [18]
H. C. Williams, Édouard Lucas and Primality Testing, Wiley (1998).