Infinitude of Sophie Germain Primes

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A prime is a positive integer greater than one whose only positive divisors are one and itself. A Sophie Germain prime is a prime $p$ for which $2p+1$ is also prime. Write \[ S(x)=\#\{p\le x:p\text{ and }2p+1\text{ are prime}\}. \]

Statement recorded in the supplied source.

Are there infinitely many Sophie Germain primes? Equivalently, is $S(x)\to\infty$ as $x\to\infty$, or does every bound $B\in\mathbb N$ admit a prime $p>B$ with $2p+1$ prime? [S2]

Short English statement

Are there infinitely many primes whose double plus one is also prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1859 (GUY-A7a). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Matteo Bordignon and Ethan Simpson Lee, Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials, arXiv:2211.11012 (2022), Section 1, the two linear forms $x,2x+1$ and Sophie Germain primes. https://arxiv.org/abs/2211.11012.