Perfect squares have at most five divisors close to its square root

TL;DR

This paper proves that perfect squares have at most five divisors within a specific small interval around their square root, addressing a conjecture by Erdős and Rosenfeld.

Contribution

It establishes a bound on the number of divisors near the square root of perfect squares, confirming a conjecture for this special case.

Findings

Every perfect square has at most five divisors close to its square root.

The result applies to divisors within a range proportional to the fourth root of the number.

Supports the conjecture of Erdős and Rosenfeld for perfect squares.

Abstract

In this paper, we consider a conjecture of Erd\H{o}s and Rosenfeld when the number is a perfect square. In particular, we show that every perfect square $n$ can have at most five divisors between $\sqrt{n} - c \sqrt[4]{n}$ and $\sqrt{n} + c \sqrt[4]{n}$.