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

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 $x/(\log x{)}^3$.

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