Residue Classes Having Tardy Totients

TL;DR

This paper demonstrates the existence of residue classes with solutions to Euler's totient congruences that grow faster than any polynomial rate, and shows infinitely many Carmichael function values in even classes.

Contribution

It provides an effective construction of residue classes with solutions to totient congruences exhibiting super-polynomial growth and proves infinite Carmichael function values in even classes.

Findings

Existence of residue classes with minimal solutions growing faster than any polynomial.

Infinitely many Carmichael function values in even residue classes.

Bound on the minimal solution size in even classes, n ≪ m^{13}.

Abstract

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as $k\to\infty$. Here, $\phi(n)$ 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 $n\ll m^{13}$.