Numbers with Four Close Factorizations

Tsz Ho Chan; Laura Holmes; Michael Liu; Jose Villarreal

paper

Numbers with Four Close Factorizations

arXiv:2508.02818

Abstract

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

20 pages, any comments are welcome