On equations $\sigma(n)=\sigma(n+k)$ and $\varphi(n)=\varphi(n+k)$

TL;DR

This paper investigates the distribution of solutions to equations involving the sum-of-divisors function and Euler's totient function, providing new upper bounds for the number of solutions within these equations.

Contribution

It introduces new upper bounds for solutions to the equations (n)=(n+k) and (n)=(n+k), advancing understanding of their distribution.

Findings

Derived new upper bounds for solutions

Improved understanding of solution distribution

Potential implications for number theory conjectures

Abstract

We study the distribution of solutions of equations $\sigma(n)=\sigma(n+k)$ and $\varphi(n)=\varphi(n+k)$. We give new upper bounds for these solutions.