Products of consecutive integers with unusual anatomy

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 Erdős 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 $a_1! a_2! a_3! = m^2$.

Figures (2)

• Figure 1: Plots of $\#({\mathcal{A}} \cap [1,x])$ for ${\mathcal{A}} = {\mathcal{B}}, {\mathcal{B}}^1, {\mathcal{V}B}, {\mathcal{F}}_3, {\mathcal{F}}_3^1$. The set ${\mathcal{V}B}^1$ is not depicted as it numerically coincides with ${\mathcal{V}B}$. (Image generated by Gemini.)
• Figure 2: For a given choice of $\frac{N}{H^2}$, the set in \ref{['valid']} is the shaded region between the blue and green lines intersected with the vertical line at $\frac{N}{H^2}$. For $\frac{N}{H^2} \geq \frac{1}{2}$, this measure is always comparable to the distance $\frac{3}{4} \frac{N}{H^2}$ between these lines. (Image generated by Gemini.)

Theorems & Definitions (63)

• Theorem 1.1
• proof
• Definition 1.2: Bad, very bad, $F_3$ intervals
• Remark 1.3
• Definition 1.4: Bad, very bad, $F_3$ sets
• Remark 1.5
• Lemma 1.6: Asymptotics for ${\mathcal{B}}^1$
• proof
• Theorem 1.7: Asymptotic for bad sets
• Theorem 1.8: Asymptotic for very bad sets