Estimating the density of a set of primes with applications to group theory

TL;DR

This paper estimates the asymptotic density of primes with specific divisor properties related to group theory, under a conjecture, showing their counts grow like a constant times x divided by (log x)^3.

Contribution

It provides the first asymptotic density estimates for primes with these divisor constraints, assuming a generalized Hardy–Littlewood conjecture.

Findings

Density of set A estimated as a constant times x / (log x)^3

Density of subset B also estimated similarly under conjecture

Results connect prime divisor properties to asymptotic prime distribution

Abstract

We estimate the asymptotic density of the set $\bar{A}$ of primes $p$ satisfying the constraint that $p+1$ and $p-1$ have only one prime divisor larger than $3$. We also estimate the density of a maximal subset $\bar{B} \subset \bar{A}$ such that for $p_1, p_2 \in \bar{B}$ no common prime divisor of $p_1(p_1 + 1)(p_1 - 1)$ and $p_2 (p_2 + 1)(p_2 - 1)$ is larger than $3$. Assuming a generalized Hardy--Littlewood conjecture, we prove that for both $\bar{A}$ and $\bar{B}$ the number of elements lesser than $x$ is asymptotically equal to a constant times $ x / (\log x)^3$.