Products of shifted primes simultaneously taking perfect power values

TL;DR

This paper establishes lower bounds on the quantity of squarefree integers up to a limit where the products of shifted primes are perfect powers, extending understanding of prime factorization properties.

Contribution

It provides new lower bounds for the count of integers with prime products that are perfect powers, including special cases with fixed numbers of prime factors.

Findings

Lower bounds for squarefree integers with prime products as perfect rth powers

Results for specific cases with exactly r prime factors

Extension of prime power product analysis to shifted primes

Abstract

Let $r \ge 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$, up to $x$, for which the products $\prod_{p \mid n} (p+a)$ (over primes $p$) are perfect $r$th powers for all $a \in A$. Also, in the cases $A = \{-1\}$ and $A = \{+1\}$, we will obtain a lower bound for the number of such $n$ with exactly $r$ distinct prime factors.