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d(n) & d(n+i)

Source: Iran TST 2023 ; Exam 2 Problem 1

March 15, 2023
number theorynumber of divisorsdivis

Problem Statement

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