← Notes on Rieg Theory

stash__2026-07-27-novak-carmichael-terminology.tex

% Reusable terminology omitted from the main text on 2026-07-27.
%
% The term is not needed for the finite-to-profinite flow of Section 3.
% Kalmynin's 2017 follow-up paper says explicitly that the author introduced
% Novák--Carmichael numbers in the 2016 preprint version of "On Novák
% numbers".  Section 5 and Theorem 6 of that paper contain the definition and
% the prime-divisor criterion below.  The published version is:
% A. B. Kalmynin, On Novák numbers, Sbornik: Mathematics 209 (2018).
%
% This file is not included by main.tex.

\begin{definition}[Nov\'ak--Carmichael number]
A positive integer $n$ is a \demph{Nov\'ak--Carmichael number} if
\[
    x^n\equiv1\mod n
\]
for every integer $x$ coprime to $n$ \cite[Section 5]{kalmynin2016novak}.
Equivalently,
\[
    p\mid n\Longrightarrow p-1\mid n
\]
for every prime $p$ \cite[Theorem 6]{kalmynin2016novak}.  If
$\lambda(n)$ denotes the Carmichael function, that is, the least positive
integer $m$ such that $x^m\equiv1\mod n$ for every integer $x$ coprime to
$n$, this is also equivalent to $\lambda(n)\mid n$
\cite{carmichael1910note}.
\end{definition}