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\chapter{Exercises}
\label{chap:Exercises}
% TODO
%
% All exercises should be group with a chapter
%
% ▶ Recommended
% M Matematically oriented
% HM Higher matematical education required
%
% 00 Immediate
% 10 Simple
% 20 Medium
% 30 Moderately hard
% 40 Term project
% 50 Research project
%
% ⁺ High risk of undershoot difficulty
\begin{enumerate}[label=\textbf{\arabic*}.]
\item {[\textit{M10}]} \textbf{Convergence of the Lucas Number ratios}
Find an approximation for $\displaystyle{ \lim_{n \to \infty} \frac{L_{n + 1}}{L_n}}$,
where $L_n$ is the $n^{\text{th}}$ Lucas Number \psecref{sec:Lucas numbers}.
\( \displaystyle{
L_n \stackrel{\text{\tiny{def}}}{\text{=}} \left \{ \begin{array}{ll}
2 & \text{if} ~ n = 0 \\
1 & \text{if} ~ n = 1 \\
L_{n - 1} + L_{n + 1} & \text{otherwise}
\end{array} \right .
}\)
\item {[\textit{M12${}^+$}]} \textbf{Factorisation of factorials}
Implement the function
\vspace{-1em}
\begin{alltt}
void factor_fact(z_t n);
\end{alltt}
\vspace{-1em}
\noindent
which prints the prime factorisation of $n!$ (the $n^{\text{th}}$ factorial).
The function shall be efficient for all $n$ where all primes $p \le n$ can
be found efficiently. You can assume that $n \ge 2$. You should not evaluate $n!$.
\item {[\textit{M20}]} \textbf{Reverse factorisation of factorials}
You should already have solved ``Factorisation of factorials''
before you solve this problem.
Implement the function
\vspace{-1em}
\begin{alltt}
void unfactor_fact(z_t x, z_t *P,
unsigned long long int *K, size_t n);
\end{alltt}
\vspace{-1em}
\noindent
which given the factorsation of $x!$ determines $x$.
The factorisation of $x!$ is
$\displaystyle{\prod_{i = 1}^{n} P_i^{K_i}}$, where
$P_i$ is \texttt{P[i - 1]} and $K_i$ is \texttt{K[i - 1]}.
\item {[\textit{05}]} \textbf{Fast primality test}
$(x + y)^p \equiv x^p + y^p ~(\text{Mod}~p)$
for all primes $p$ and for a few composites $p$.
Use this to implement a fast primality tester.
\end{enumerate}
\chapter{Solutions}
\label{chap:Solutions}
\begin{enumerate}[label=\textbf{\arabic*}.]
\item \textbf{Convergence of the Lucas Number ratios}
It would be a mistake to use bignum, and bigint in particular,
to solve this problem. Good old mathematics is a much better solution.
$$ \lim_{n \to \infty} \frac{L_{n + 1}}{L_n} = \lim_{n \to \infty} \frac{L_{n}}{L_{n - 1}} = \lim_{n \to \infty} \frac{L_{n - 1}}{L_{n - 2}} $$
$$ \frac{L_{n}}{L_{n - 1}} = \frac{L_{n - 1}}{L_{n - 2}} $$
$$ \frac{L_{n - 1} + L_{n - 2}}{L_{n - 1}} = \frac{L_{n - 1}}{L_{n - 2}} $$
$$ 1 + \frac{L_{n - 2}}{L_{n - 1}} = \frac{L_{n - 1}}{L_{n - 2}} $$
$$ 1 + \varphi = \frac{1}{\varphi} $$
So the ratio tends toward the golden ratio.
\item \textbf{Factorisation of factorials}
Base your implementation on
\( \displaystyle{
n! = \prod_{p~\in~\textbf{P}}^{n} p^{k_p}, ~\text{where}~
k_p = \sum_{a = 1}^{\lfloor \log_p n \rfloor} \lfloor np^{-a} \rfloor.
}\)
There is no need to calculate $\lfloor \log_p n \rfloor$,
you will see when $p^a > n$.
\item \textbf{Reverse factorisation of factorials}
$\displaystyle{x = \max_{p ~\in~ P} ~ p \cdot f(p, k_p)}$,
where $k_p$ is the power of $p$ in the factorisation
of $x!$. $f(p, k)$ is defined as:
\vspace{1em}
\hspace{-2.8ex}
\begin{minipage}{\linewidth}
\begin{algorithmic}
\STATE $k^\prime \gets 0$
\WHILE{$k > 0$}
\STATE $a \gets 0$
\WHILE{$p^a \le k$}
\STATE $k \gets k - p^a$
\STATE $a \gets a + 1$
\ENDWHILE
\STATE $k^\prime \gets k^\prime + p^{a - 1}$
\ENDWHILE
\RETURN $k^\prime$
\end{algorithmic}
\end{minipage}
\vspace{1em}
\item \textbf{Fast primality test}
If we select $x = y = 1$ we get $2^p \equiv 2 ~(\text{Mod}~p)$. This gives us
\vspace{-1em}
\begin{alltt}
enum zprimality ptest_fast(z_t p)
\{
z_t a;
int c = zcmpu(p, 2);
if (c <= 0)
return c ? NONPRIME : PRIME;
zinit(a);
zsetu(a, 1);
zlsh(a, a, p);
zmod(a, a, p);
c = zcmpu(a, 2);
zfree(a);
return c ? NONPRIME : PROBABLY_PRIME;
\}
\end{alltt}
\end{enumerate}
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