Finding Almost Squares III

TL;DR

This paper investigates the distribution of integers with multiple factorizations near the square root, improving bounds on short intervals containing such numbers and exploring related divisibility properties.

Contribution

It advances previous results by providing tighter bounds on short intervals containing almost squares of type 2 and examines divisibility within specific ranges.

Findings

Improved bounds on short intervals containing almost squares of type 2

Enhanced understanding of divisibility properties in short intervals

New results on the distribution of integers with multiple factorizations

Abstract

An almost square of type 2 is an integer $n$ that can be factored in two different ways as $n = a_1 b_1 = a_2 b_2$ with $a_1$, $a_2$, $b_1$, $b_2 \approx \sqrt{n}$. In this paper, we shall improve upon previous result on short intervals containing an almost square of type 2. This leads to an inquiry of finding a short interval around $x$ that contains an integer divisible by some integer in $[x^c, 2 x^c]$ with $0 < c < 1$.