How strong can primes be
Volume 179 / 2017
Acta Arithmetica 179 (2017), 363-373
MSC: 11Y05, 11N05, 11N13.
DOI: 10.4064/aa8578-5-2017
Published online: 14 June 2017
Abstract
We prove that there are a positive proportion of primes $p$ such that $p+1$ has a prime factor at least $\sqrt{p}$, $p-1$ has a prime factor $q$ at least $\sqrt{p}$, and $q-1$ has a prime factor at least $p^{0.0705}$. Moreover, there are a positive proportion of primes $p$ such that both $p+1$ and $p-1$ have prime factors at least $p^\theta$ with $\theta={1}/{2}+{1}/{36}.$ These are related to strong primes appearing in RSA schemes.
