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arXiv · 1810.08679

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

Carlos Esparza·Lukas Gehring
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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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Carlos Esparza, Lukas Gehring. 2018-10-19. Estimating the density of a set of primes with applications to group theory. https://arxiv.org/abs/1810.08679

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