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?

  • $\begingroup$ $5$ isn't...... $\endgroup$
    – Randall
    Aug 2 at 14:34
  • $\begingroup$ $2$ isn't........ $\endgroup$ Aug 2 at 14:34
  • $\begingroup$ I suppose lots of them aren't. Did you do an internet search? en.wikipedia.org/wiki/Mersenne_prime $\endgroup$
    – Randall
    Aug 2 at 14:35
  • $\begingroup$ Of course not, but the largest known are Mersenne and the most other of the largest generalized Fermat primes since they are easiest to be proven prime and moreover give a better chance to find a prime than if one would take a random number. $\endgroup$
    – Peter
    Aug 2 at 14:36
  • $\begingroup$ Sorry, I assumed 2, 3, and 5 were prime and started calculating from these. $\endgroup$
    – Matt
    Aug 2 at 14:38

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 $11.$


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