*Speaker:* Dr John Cosgrave (http://staff.spd.dcu.ie/johnbcos/)

*Title:* Extensions of the Gauss-Wilson theorem

*Time:* 5:00PM

*Date:* Wed 26th March 2008

*Location: *Mathematical Sciences Seminar Room

*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.

(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)