Residue Classes Having Tardy Totients

Abstract

We show, in an effective way, that there exists a sequence of congruence classes $a_k(\mathrm{mod}{m}_k)$ such that the minimal solution $n=n_k$ of the congruence $\varphi (n)\equiv a_k(\mathrm{mod}{m}_k)$ exists and satisfies $\log n_k/\log m_k\to \infty$ as $k\to \infty$. Here, $\varphi (n)$ is the Euler function. This answers a question raised in \cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $\lambda (n)$ and the least such $n$ satisfies $n\ll m^{13}$.

14 pages

Cited by in corpus (2)

• Distribution in coprime residue classes of polynomially-defined multiplicative functions
• Smallest totient in a residue class