Sponsored by the Rutgers University Department of Mathematics and the

Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

**Co-organizers:****Andrew Baxter**, Rutgers University, baxter{at} math [dot] rutgers [dot] edu**Lara Pudwell**, Rutgers University, lpudwell {at} math [dot] rutgers [dot] edu**Doron Zeilberger**, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Extensions of the Gauss-Wilson theorem

Speaker: **John Cosgrave**, Dublin, Ireland

Date: Thursday, April 10, 2008 5:00pm

Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Karl Dilcher and I have made the first extension of the G-W theorem since the appearance of Gauss' Disquisitiones. Defining N_n! - the 'Gauss factorial' of N with respect to n - to be the product of the residue classes in [1, N] that are relatively prime to n, we have given a complete determination of the order of (n-1/2)_n! mod n. This is a composite modulus extension of Mordell's 1961 result concerning the order of (p-1/2)! mod p (prime p).

I will outline work-in-progress concerning the order of (n-1/M)_n! mod n for M = 3 and 4, introduce a new class of primes (Gauss-4 primes), and outline a number of open problems.

Maple has played a vital role in these investigations.