On the Tower Factorization of Integers

Abstract

Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers p^a; factoring each exponent a as a product of prime powers q^b, and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of “floors” in its tower factorization. In particular, given a fixed integer $k\ge 1$, we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.

MSC:

• 11A25
• 11A41
• 11A51

Acknowledgment

The authors wish to thank the referees and the editor for their very helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Jean-Marie De Koninck

JEAN-MARIE DE KONINCK obtained his Ph.D. in mathematics from Temple University in 1972 under the supervision of Emil Grosswald. After 44 years as a faculty member at Université Laval in Quebec, Canada, he retired in 2016 and is now Professor Emeritus. His main research interest is the multiplicative structure of integers. He is still involved in math outreach and takes pleasure in swimming and writing books. His latest, coauthored with Nicolas Doyon, is The Life of Primes in 37 Episodes.

Département de mathématiques et de statistique, Université Laval, Québec G1V 0A6, Canada

[email protected]

William Verreault

WILLIAM VERREAULT is a master’s student at Université Laval and a soon-to-be Vanier Scholar at the University of Toronto. While his main research interests lie in analysis, he enjoys working on various number theory problems.

Département de mathématiques et de statistique, Université Laval, Québec G1V 0A6, Canada

[email protected]