Sophie Germain prime
If a Sophie Germain prime "p" is congruent to 3 (mod 4) (, "Lucasian primes"), then its matching safe prime 2"p" + 1 will be a divisor of the Mersenne number 2"p" − 1. Historically, this result of Leonhard Euler was the first known criterion for a Mersenne number with a prime index to be composite. It can be used to generate the largest Mersenne numbers (with prime indices) that are known to be composite.