Extensions of the Gauss-Wilson theorem
Speaker: Dr John Cosgrave (http://staff.spd.dcu.ie/johnbcos/)
Date: Wed 26th March 2008
Location: Mathematical Sciences Seminar Room
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.)