Sorry if this is a duplicate of a post, but I having trouble finding the answer anywhere. Are all prime numbers Mersenne prime? Or are there any numbers in between Mersenne primes?
All Mersenne primes are prime numbers, but the converse is not necessarily true. A Mersenne prime is $2^k - 1$, where $k$ is a positive integer greater than or equal to 2. Examples of Mersenne primes are $3$, $7$, and $31$. These are all also regular primes. However, counterexamples of primes that are NOT Mersenne primes are (as shown in the comments), $2$, $5$, or $11.$