Infinitude of Sophie Germain primes

Determine whether there are infinitely many primes p such that 2p + 1 is also prime (i.e., whether there are infinitely many Sophie Germain primes).

Background

In the historical timeline, the authors recall Sophie Germain’s criterion: if p is a prime with 2p + 1 also prime, then Fermat’s Last Theorem holds for exponent p under certain coprimality conditions. Despite extensive computational evidence, the fundamental question of whether infinitely many such primes exist remains unresolved.

This question is central in analytic number theory and relates to the distribution of primes in specific linear forms; it is independent of Fermat’s Last Theorem, which is now proved, but persists as a prominent open problem in prime number theory.