# Gaps between divisible terms in $a^2 (a^2 + 1)$

## TL;DR

This paper investigates the divisibility properties of the sequence defined by a^2(a^2+1), establishing a lower bound on the gap between terms and exploring implications under the abc conjecture.

## Contribution

It improves previous results by establishing a new gap principle for divisibility in the sequence without extra assumptions and under the abc conjecture.

## Key findings

- Proves a new lower bound on the gap between divisible terms.
- Establishes a conditional bound under the abc conjecture.
- Advances understanding of divisibility gaps in quadratic sequences.

## Abstract

Suppose $a^2 (a^2 + 1)$ divides $b^2 (b^2 + 1)$ with $b > a$. In this paper, we improve a previous result and prove a gap principle, without any additional assumptions, namely $b \gg a (\log a)^{1/8} / (\log \log a)^{12}$. We also obtain $b \gg_\epsilon a^{15/14 - \epsilon}$ under the abc conjecture.