Kevin Ford, Sergei Konyagin
Published 2020-07-11Version 1
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.
On the Lehmer's problem involving Euler's totient function
Solutions of $φ(n)=φ(n+k)$ and $σ(n)=σ(n+k)$
Sparse subsets of the natural numbers and Euler's totient function
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