The set of values of any finite iteration of Euler's \(\varphi\) function contains long arithmetic progressions (Q6595591)

A prime number \(p\) is a fixed prime divisor of a polynomial \(G\) if for all \( t \in \mathbb{Z} : p | G(t)\). Dickson's conjecture, which is a predecessor of the Hardy-Littlewood conjecture on prime \(k\)-tuples and also Schinzel's Hypothesis H, reads as follows:\N\NLet \(s\) be a positive integer and \(F_1, \ldots, F_s\) be linear polynomials with integral coefficients and positive leading coefficient such that their product has no fixed prime divisor. Then, there exist infinitely many positive integers \(n\) such that \(F_1(n), \ldots, F_s(n)\) are all primes.\N\NThe only case where Dickson's conjecture is known to be true is for \(s = 1\), which coincides with the Dirichlet's theorem on the primes in arithmetic progressions.\N\NIn the paper under review, the authors prove that assuming Dickson's conjecture is true and letting \(a\geq 2\) be a positive integer, there exists a positive integer \(D_a\) such that for any positive integer \(H\) there exist positive integers \(M, m_1, \ldots , m_H\) such that for all \(h\) in \([1, H]\),\N\[\N\varphi^{(a)}(m_h)=D_ah+M,\N\]\Nwhere \(\varphi^{(a)}=\varphi\circ\cdots\circ\varphi\) \((a\,\text{times})\) is iterated Euler's totient. The case \(a=1\) has been studied already by \textit{J.-M. Deshouillers} et al. in [Acta Arith. 199, No. 1, 103--109 (2021; Zbl 1487.11010)].