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

2018/10/19 by Carlos Gamero Esparza, Lukas Gehring, Esparza, Carlos +1
Mathematics · #11N05 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1810.08679

openalex publication_date 2018/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We estimate the asymptotic density of the set 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 B ⊂ A such that for p₁, p₂ ∈ B no common prime divisor of p₁(p₁ + 1)(p₁ - 1) and p₂ (p₂ + 1)(p₂ - 1) is larger than 3. Assuming a generalized Hardy--Littlewood conjecture, we prove that for both A and B the number of elements lesser than x is asymptotically equal to a constant times x / (log x)³.

Related

• Roth's Theorem and the Hardy--Littlewood majorant problem for thin subsets of primes
• Very small intervals containing at least three primes
• Sums of singular series with large sets and the tail of the distribution of primes
• Asymptotic of the greatest distance between adjacent primes and the Hardy-Littlewood conjecture
• Asymptotics of the number of primes and sums of functions of primes in a subset of natural series