Mersenne number ($M_k=2^k-1,k\ge 1$)

A Mersenne number has the form $M_k=2^k-1$ for a positive integer $k$. Its binary representation consists of $k$ ones. For example, $M_4=15$ is composite, while $M_5=31$ is prime. If $k=ab$ with $a,b>1$, the finite geometric series factorizationhas two factors greater than one. Consequently a Mersenne prime requires a prime exponent, but $M_{11}=23\cdot 89$ shows that this condition alone is insufficient.

A Mersenne prime is a Mersenne number that is a prime number. It is the prime factor required by the Euclid-Euler theorem; $3,7,31,127$ generate $6,28,496,8128$. The family is named after Marin Mersenne.