Infinitude of Sophie Germain primes

As of 2026, the infinitude of Sophie Germain primes remains completely open. Like the twin prime conjecture, the problem asks for simultaneous prime values of two linear forms ($n$ and $2n+1$). Weighted sieve methods prove that there are infinitely many primes $p$ such that $2p+1$ is an almost-prime with at most two prime factors ($P_2$), and that the sum of reciprocals of Sophie Germain primes converges. However, the sieve-theoretic parity barrier prevents existing methods from distinguishing primes from products of two primes in $2p+1$. Furthermore, because the linear forms $n$ and $2n+1$ have different leading coefficients, the Maynard–Tao bounded-gaps machinery does not directly yield pairs of primes of the exact form $(p, 2p+1)$.