Characterizations of Mersenne and 2-Rooted Primes
We give several characterizations of Mersenne primes and of primes for which 2 is a primitive root. These characterizations involve group algebras, circulant matrices, binomial coefficients, and bipartite graphs.
Chebolu, Sunil K., Keir Lockridge, and Gaywalee Yamskulna. "Characterizations of Mersenne and 2-Rooted Primes." Finite Fields and Their Applications 35 (September 2015), 330-351.