The Most Frequent Values of the Largest Prime Divisor Function

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.

KEYWORDS:

• largest prime divisor
• smooth numbers
• factoring algorithms

2000 AMS Subject Classification:

• 11N37

Acknowledgments

The work contained in this article comprises a portion of my Ph.D. thesis [McNew 15]. I would like to thank my advisor, Carl Pomerance, for his invaluable guidance and encouragement throughout the development of this article. I would also like to thank Florian Luca for listening to early versions of the arguments presented here, Jean-Marie De Koninck for pointing me to his work related to my problem, Sary Drappeau for suggesting the use of Saias’ work to improve Theorem 4.1, Jonathan Bober for discussions about computing the popular primes, Tomás Oliveira e Silva for pointing out a mistake in the computations presented in an earlier draft of the article, and Robin Pemantle for his interest in this project.