Numbers with Four Close Factorizations

TL;DR

This paper investigates numbers with four close factorizations, establishing an optimal upper bound for the factorization parameter and transforming the problem into generalized Pell equations to analyze solutions.

Contribution

It introduces a novel approach by converting the factorization problem into generalized Pell equations and derives the best possible upper bound for the parameter A.

Findings

Established the upper bound A ≤ 0.04742...·C^3 + O(C)

Connected factorization properties to solutions of Pell equations

Provided a method to analyze numbers with multiple close factorizations

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 $B \le A$, $1 \le a_1 < a_2 < a_3 \le C$ and $1 \le b_1 < b_2 < b_3 \le C$. We obtain the optimal upper bound $A \le 0.04742 \ldots \cdot C^3 + O(C)$. The key idea is to transform the original question into generalized Pell equations $a x^2 - b y^2 = c$ and study their solutions.