Numbers with Four Close Factorizations

In this paper, we study numbers $n$ that can be factored in four different ways as $n=AB=(A+a_1)(B-b_1)=(A+a_2)(B-b_2)=(A+a_3)(B-b_3)$ with $B\le A$, $1\le a_1<a_2<a_3\le C$ and $1\le b_1<b_2<b_3\le C$. We obtain the optimal upper bound $A\le 0.04742\dots \cdot C^3+O(C)$. The key idea is to transform the original question into generalized Pell equations $a{x}^2-b{y}^2=c$ and study their solutions.