Gaps between totients

TL;DR

This paper investigates the set of integers for which the difference of Euler's totient function values has infinitely many solutions, revealing structural properties and connections to deep conjectures in number theory.

Contribution

It establishes bounds on the minimal elements of D, identifies infinite subsets within certain arithmetic progressions, and links the structure of D to the Generalized Elliott-Halberstam Conjecture.

Findings

Minimum of D is at most 154

Every multiple of a specific A is in D

Certain arithmetic progressions contain infinitely many elements of D

Abstract

We study the set D of positive integers d for which the equation $\phi(a)-\phi(b)=d$ has infinitely many solution pairs (a,b), where $\phi$ is Euler's totient function. We show that the minumum of D is at most 154, exhibit a specific A so that every multiple of A is in D, and show that any progression a mod d with 4|a and 4|d, contains infinitely many elements of D. We also show that the Generalized Elliott-Halberstam Conjecture, as defined in [6], implies that D equals the set of all positive, even integers.