Differences between totients
Volume 200 / 2021
Acta Arithmetica 200 (2021), 61-69
MSC: Primary 11A25, 11N64; Secondary 11D85.
DOI: 10.4064/aa200711-19-4
Published online: 2 September 2021
Abstract
We study the set $\mathcal D $ of positive integers $d$ for which the equation $\phi (a)-\phi (b)=d$ has infinitely many solution pairs $(a,b)$. We show that $\min \mathcal D \le 154$, exhibit a specific $A$ such that every multiple of $A$ is in $\mathcal D $, and show that any progression $a\mod d$ with $4\,|\, a$ and $4\,|\, d$ contains infinitely many elements of $\mathcal D $. We also show that the Generalized Elliott–Halberstam Conjecture, as defined by Polymath (2014), implies that $\mathcal D $ contains all positive, even integers.
