Density of Symmetric 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 symmetric pair is a pair of distinct odd primes $p,q$ with $\gcd(p-1,q-1)=|p-q|$. A prime is symmetric if it belongs to at least one such pair. Its partner is not required to lie below the counting threshold. Equivalently, if $S(q,p)$ counts integer points $(m,n)$ with $0<m<p/2$, $0<n<q/2$ and $n<(q/p)m$, then the pair is symmetric when $S(q,p)=S(p,q)$. Define \[ P(x)=\left|\{p<x:p\text{ is an odd prime and }\exists q\ne p\text{ an odd prime with }\gcd(p-1,q-1)=|p-q|\}\right|. \] All logarithms are natural, and $\sigma=2-(1+\log\log2)/\log2$.

Statement recorded in the supplied source.

Is \[ P(x)=\frac{x}{(\log x)^{\sigma+o(1)}}\qquad(x\to\infty)? \] Equivalently, does $\left(\log x-\log P(x)\right)/\log\log x$ tend to $\sigma$? This is a counting question about primes belonging to pairs, rather than the number of pairs whose two members are below $x$. [S2]

Short English statement

Does the number of primes below x that have a symmetric-prime partner have the proposed logarithmic exponent?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1874 (GUY-A20). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 William Banks, Paul Pollack and Carl Pomerance, Symmetric primes revisited, arXiv:1908.06161v2 (2019), §1, Theorems 1.1--1.2 and the density conjecture following them. https://arxiv.org/abs/1908.06161v2.