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.

References

Note that to this day it is still not known if there are infinitely many such primes.

— Understanding Fermat's Last Theorem's Proofs  (2508.10362 - Qiu et al., 14 Aug 2025) in Section 2.1 (Classical Contributions)

It is conjectured that there are infinitely many safe primes, see for example , which would make the result above hold for infinitely many $n$.

— On exceptional cliques in matrix rings  (2608.24586 - Boutros et al., 25 Aug 2026) in Introduction