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	Add exercise: [05] Fast primality test - libzahl - big integer library
	git clone git://git.suckless.org/libzahl
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	---
	commit ccc36c882dc899ce75e41c7675ce48263ad24bfa
	parent 555b57b3190c2ed6f73970c0515ac77dc4087220
	Author: Mattias Andrée <[email protected]>
	Date: Sun, 24 Jul 2016 03:58:08 +0200

	Add exercise: [05] Fast primality test

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/exercises.tex \| 37 +++++++++++++++++++++++++++++…

	1 file changed, 37 insertions(+), 0 deletions(-)
	---
	diff --git a/doc/exercises.tex b/doc/exercises.tex
	@@ -36,6 +36,7 @@ where $L_n$ is the $n^{\text{th}}$ Lucas Number \psecref{sec:…
	}\)


	+
	\item {[\textit{M12${}^+$}]} \textbf{Factorisation of factorials}

	Implement the function
	@@ -52,6 +53,7 @@ The function shall be efficient for all $n$ where all primes …
	be found efficiently. You can assume that $n \ge 2$. You should not evaluate $…


	+
	\item {[\textit{M20}]} \textbf{Reverse factorisation of factorials}

	You should already have solved ``Factorisation of factorials''
	@@ -73,6 +75,15 @@ $\displaystyle{\prod_{i = 1}^{n} P_i^{K_i}}$, 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}


	@@ -101,6 +112,7 @@ $$ 1 + \varphi = \frac{1}{\varphi} $$
	So the ratio tends toward the golden ratio.


	+
	\item \textbf{Factorisation of factorials}

	Base your implementation on
	@@ -114,6 +126,7 @@ There is no need to calculate $\lfloor \log_p n \rfloor$,
	you will see when $p^a > n$.


	+
	\item \textbf{Reverse factorisation of factorials}

	$\displaystyle{x = \max_{p ~\in~ P} ~ p \cdot f(p, k_p)}$,
	@@ -140,4 +153,28 @@ of $x!$. $f(p, k)$ is defined as:



	+\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}