The set of values of any finite iteration of Euler’s $\varphi $ function contains long arithmetic progressions
Volume 214 / 2024
Acta Arithmetica 214 (2024), 343-351
MSC: Primary 11B83; Secondary 11B05, 11N32, 11N64
DOI: 10.4064/aa230601-7-9
Published online: 24 January 2024
Abstract
Assuming the validity of Dickson’s conjecture, we show that the set of values of iterated Euler’s totient $\varphi $ function $\varphi \circ \cdots \circ \varphi $ ($n$ times) contains arbitrarily long arithmetic progressions with an explicitly given common difference $D_a$ depending only on $a$. This extends a previous result (case $a = 1$) of Deshouillers, Eyyunni and Gun. In particular, this implies that this set has upper Banach density at least $1/D_a \gt 0$.
