Characterizations of Mersenne and 2-rooted primes

Abstract

We give several characterizations of Mersenne primes (Theorem 1.1) and of primes for which 2 is a primitive root (Theorem 1.2). These characterizations involve group algebras, circulant matrices, binomial coefficients, and bipartite graphs.

19 pages, final version, to appear in Finite Fields and their Applications