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 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.
