I was reading about Mersenne primes and wondering about its connection with the Goldbach conjecture. It's conjectured that there's infinitely many Mersenne primes.
One direction is trivial, that the even integer $2^{n+1} - 2$ can be written as the sum of two (Mersenne) primes. The converse, which is the GC, might not be true. Still, I want to ask what relationship between Mersenne primes and the GC is, and if it's any significant.