On shifted primes with large prime factors and their products

Abstract

We estimate from below the lower density of the set of prime numbers $p$ such that $p-1$ has a prime factor of size at least $p^c$, where $1/4 \le c \leq 1/2$. We also establish upper and lower bounds on the counting function of the set of positive integers $n\le x$ with exactly $k$ prime factors, counted with or without multiplicity, such that the largest prime factor of ${\text{\rm gcd}}(p-1: p\mid n)$ exceeds $n^{1/2k}$.