The equation $\omega(n)=\omega(n+1)$

TL;DR

This paper proves that there are infinitely many integers n for which n and n+1 share the same number of distinct prime factors, contributing to understanding prime divisor patterns.

Contribution

It establishes the infinitude of integers with equal numbers of distinct prime divisors for consecutive numbers, a new result in number theory.

Findings

Infinitely many such n exist.

Consecutive integers can have identical counts of distinct prime factors.

Abstract

We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.