By forming a table of sequential odd numbers using a single quadrant expanding square pattern it was observed that multiple Mersenne primes fall into the first column. We prove that the Mersenne primes which fall into the first column will be of the form p = 3 mod 4, where p is the Mersenne prime exponent. Focusing the search of primes to those which fall into the first column of this square may be a means to increase the speed at which large primes are discovered.
Daignault, Cory P.; Sprague, Douglas; and Mott, Joe L.
"Evaluating Mersenne Primes Using a Single Quadrant Expanding Square,"
The Mathematics Enthusiast: Vol. 9
, Article 11.
Available at: http://scholarworks.umt.edu/tme/vol9/iss3/11