The Diophantine equation $b (b+1) (b+2) = t a (a + 1) (a + 2)$ and gap principle

TL;DR

This paper investigates divisibility relations between products of three consecutive integers and establishes lower bounds on their gaps, using advanced number theory techniques and extending to other polynomial sequences.

Contribution

It proves new lower bounds on the gaps between such products when divisibility occurs, applying the effective Liouville-Baker-Feldman theorem to these Diophantine equations.

Findings

Established a lower bound for gaps: $b \,\gg\; \frac{a \log a)^{1/6}}{(\log \log a)^{1/3}}$.

Extended the analysis to polynomial sequences like $a^2 (a^2 + l)$ dividing $b^2 (b^2 + l)$.

Demonstrated the applicability of the effective Liouville-Baker-Feldman theorem to these problems.

Abstract

In this article, we are interested in whether a product of three consecutive integers $a (a+1) (a+2)$ divides another such product $b (b+1) (b+2)$. If this happens, we prove that there is some gaps between them, namely $b \gg \frac{a \log a)^{1/6}}{\log \log a)^{1/3}}$. We also consider other polynomial sequences such as $a^2 (a^2 + l)$ dividing $b^2 (b^2 + l)$ for some fixed integer $l$. Our method is based on effective Liouville-Baker-Feldman theorem.