Products of consecutive integers with unusual anatomy

TL;DR

This paper investigates special properties of intervals of consecutive integers related to divisibility and factorial equations, providing asymptotic estimates for their occurrences and solutions.

Contribution

It introduces new classifications of intervals based on divisibility properties and derives asymptotic formulas for their counts and related factorial equation solutions.

Findings

Asymptotics for the number of bad and very bad intervals

Near-asymptotics for type F3 interval endpoints

Results on solutions to the factorial equation a1! a2! a3! = m^2

Abstract

Call an interval $\{N+1,\dots,N+H\}$ of consecutive natural numbers \emph{bad} if the product $(N+1) \dots (N+H)$ is divisible by the square of its largest prime factor; \emph{very bad} if this product is powerful, and \emph{type $F_3$} if it has the same squarefree component as a factorial. Such concepts arose in the analysis of the factorial equation $a_1! a_2! a_3! = m^2$ with $a_1<a_2<a_3$. Answering several questions of Erd\H{o}s and Graham, we obtain asymptotics for the number of integers contained in bad or very bad intervals, and to get near-asymptotics for the number of right endpoints of a type $F_3$ interval, or on the number of solutions to $a_1! a_2! a_3! = m^2$.