Popular values of the largest prime divisor function

TL;DR

This paper studies the distribution of the largest prime divisor of integers in [2,x], identifying the most frequent primes (modes) and analyzing their properties, with applications to factoring algorithms.

Contribution

It provides an asymptotic formula for the mode of the largest prime divisor distribution and characterizes the set of 'popular primes' that frequently appear as the largest prime divisor.

Findings

Many primes never appear as the mode of the largest prime divisor.

The set of 'popular primes' is characterized and compared to other prime subsets.

Techniques are applied to analyze a problem in factoring algorithms.

Abstract

We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as $x$ tends to infinity, we look at the set of those prime numbers which, for some value of $x$, occur most frequently as the largest prime divisor of the integers in the interval $[2,x]$. We find that many prime numbers never have this property. We compare the set of "popular primes," those primes which are at some point the mode, to other interesting subsets of the prime numbers. Finally, we apply the techniques developed to a similar problem which arises in the analysis of factoring algorithms.