Problem

Source: Iran TST 2023 ; Exam 2 Problem 1

Tags: number theory, number of divisors, divis



Suppose that $d(n)$ is the number of positive divisors of natural number $n$. Prove that there is a natural number $n$ such that $$ \forall i\in \mathbb{N} , i \le 1402: \frac{d(n)}{d(n \pm i)} >1401 $$ Proposed by Navid Safaei and Mohammadamin Sharifi