It is a clear that largest known primes are Mersenne prime. It is well known that $2^p - 1$ is prime only if $p$ is prime; however, the converse is not true - take $p = 11$. My question is: is there some nontrivial characterization of which primes $p$ have the property that $2^p - 1$ is prime? I think from GIMP theorist up to some exceptional properties .


**Note**: Actually I find this property "The only exponent of known largest prime equal to $3 \bmod 11$ is $77232917$



What are the exceptional properties of Mersenne exponent for known largest prime?
Primes $p$ such that $2^p-1$ is prime are called *Mersenne exponents*. You can find many of their properties on the corresponding entry of the OEIS, namely [A000043](https://oeis.org/A000043). Beyond their obvious connections with perfect numbers, there are also characterisations like 



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> The (prime) number $p$ appears in this sequence if and only if there is no prime $q<2^p-1$ such that the order of $2$ modulo $q$ equals $p$.
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As a particular case of this, we have



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> If $p=4k+3$ is prime and also $q=2p+1$ is prime then the order of $2$ modulo $q$ is $p$, so $p$ is not a term of this sequence.
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There is also the following characterisation.



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> $p$ is a Mersenne exponent if and only if every degree $p$ irreducible polynomial over GF(2) is also primitive (i.e. has order $2^p-1$).
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Some of the properties you'll find on the OEIS are often quite quick to prove. 


