% Reusable material removed from Section 3.4 on 2026-07-27.
%
% Section 3.3 now uses only the adjunction
%     lambda(a) | b  iff  a | j(b).
% The explicit formula, examples, and cofinality lemma below are not needed
% for the total-exponentiation and fixed-point argument.
%
% This file is not included by main.tex.  It preserves the old source for
% later study.  Its large numerical example has not been reverified in the
% present edit.

\begin{para}{Explicit construction of $\j$.}
    Let $n\in \Np$ be a positive integer, and
    for a prime $p$, let $v_p(n)$ be the largest integer $e\geq0$ such
    that $p^e\mid n$.
    Let $p_0<\dots <p_{k} $ be the list of all prime numbers in the set $\{d+1\mid d\text{ is a divisor of } n\}$.
    Notice that $p_0$ is always equal to $2$. Then, $\j(n)$ is given by
    \[
    \j(n) =
    p_1^{v_{p_1}(n)+1}\times \dots \times p_k^{v_{p_k}(n)+1} \times
    \begin{cases}
        2 & (\text{$n$ is odd})\\
        2^{v_2(n) +2} & (\text{$n$ is even}).
    \end{cases}
    \]
\end{para}

\begin{lemma}\label{lem:JumpCharacterizationOfShiftedCarmichael}
    For a positive integer $n\in \Np$, the following conditions are equivalent:
    \begin{itemize}
        \item For every prime divisor $p$ of $n$, the number $p-1$ divides $n$.
        \item $\lambda(n) \mid n$.
        \item For any $x\in (\Z/n\Z)^\times$, $x^{n}\equiv 1 \mod n$.
        \item $n\mid \j(n)$.
    \end{itemize}
\end{lemma}

\begin{example}[Values of $\j(n)$]
The first values of the function $\j(n)$ are given as follows:
    \begin{itemize}
        \item $\j(1) = 2$.
        \item $\j(2) = 2^3\times 3 = 24$.
        \item $\j(3) = 2$.
        \item $\j(4) = 2^4\times 3\times 5= 240$.
        \item $\j(5) = 2$.
        \item $\j(6) = 2^3 \times 3^2 \times 7 = 504$.
        \item $\j(24) = \j^3(1)= 2^5\times 3^2 \times 5\times 7\times 13 = 131040$.
        \item $\j(131040) = \j^4(1)=
        2^7 \times 3^3 \times 5^2 \times 7^2 \times 11 \times 13^2 \times 17 \times 19 \times 29 \times 31 \times 37 \times 41 \times 43 \times 53 \times 61 \times 71 \times 73 \times 79 \times 97 \times 113 \times 127 \times 131 \times 157 \times 181 \times 211 \times 241 \times 281 \times 313 \times 337 \times 421 \times 521 \times 547 \times 631 \times 673 \times 911 \times 937 \times 1009 \times 1093 \times 1171 \times 1249 \times 1873 \times 2017 \times 2081 \times 2341 \times 2521 \times 2731 \times 3121 \times 3361 \times 6553 \times 8191 \times 8737 \times 14561 \times 16381 \times 21841 \times 26209 \times 65521 \times 131041$.
    \end{itemize}
\end{example}

\begin{lemma}
    For any $n,m\in \Np$, there exists a natural number $k\in \N$ such that $n\mid \j^k(m)$.
\end{lemma}
\begin{proof}
    By the adjointness $\lambda \dashv \j$, we have the equivalence
    \[
    n\mid \j^k(m) \iff \lambda^k(n) \mid m.
    \]
    Therefore, it is enough to prove that the sequence
    \[
    n \geq \lambda(n)\geq \lambda^2(n)\geq \dots
    \]
    contains $1$. This follows since $\lambda(n)<n$ for any $n\geq 2$.
\end{proof}