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

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.