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	Add exercise: [M13] The totient from factorisation - libzahl - big integer libr…
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	commit 87e84a9167666022bba7c73b5447791bf9f6797b
	parent 18a0e23b2f2e37c56c4f0f9a3dd56c1c619a4468
	Author: Mattias Andrée <[email protected]>
	Date: Fri, 21 Oct 2016 05:20:55 +0200

	Add exercise: [M13] The totient from factorisation

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

	Diffstat:
	M doc/exercises.tex \| 57 +++++++++++++++++++++++++++++…

	1 file changed, 57 insertions(+), 0 deletions(-)
	---
	diff --git a/doc/exercises.tex b/doc/exercises.tex
	@@ -271,6 +271,38 @@ and $\varphi(1) = 1$.



	+\item {[\textit{M13}]} \textbf{The totient from factorisation}
	+
	+Implement the function
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void totient_fact(z_t t, z_t *P,
	+ unsigned long long int *K, size_t n);
	+\end{alltt}
	+\vspace{-1em}
	+
	+\noindent
	+which calculates the totient $t = \varphi(n)$, where
	+$n = \displaystyle{\prod_{i = 1}^n P_i^{K_i}} > 0$,
	+and $P_i = \texttt{P[i - 1]} \in \textbf{P}$,
	+$K_i = \texttt{K[i - 1]} \ge 1$. All values \texttt{P}.
	+\texttt{P} and \texttt{K} make up the prime factorisation
	+of $n$.
	+
	+You can use the following rules:
	+
	+\( \displaystyle{
	+ \begin{array}{ll}
	+ \varphi(1) = 1 & \\
	+ \varphi(p) = p - 1 & \text{if } p \in \textbf{P} \\
	+ \varphi(nm) = \varphi(n)\varphi(m) & \text{if } \gcd(n, m) = 1 \\
	+ n^a\varphi(n) = \varphi(n^{a + 1}) &
	+ \end{array}
	+}\)
	+
	+
	+
	\item {[\textit{HMP32}]} \textbf{Modular tetration}

	Implement the function
	@@ -711,6 +743,31 @@ then, $\varphi(n) = \varphi\|n\|$.



	+\item \textbf{The totient from factorisation}
	+
	+\vspace{-1em}
	+\begin{alltt}
	+void
	+totient_fact(z_t t, z_t *P,
	+ unsigned long long *K, size_t n)
	+\{
	+ z_t a, one;
	+ zinit(a), zinit(one);
	+ zseti(t, 1);
	+ zseti(one, 1);
	+ while (n--) \{
	+ zpowu(a, P[n], K[n] - 1);
	+ zmul(t, t, a);
	+ zsub(a, P[n], one);
	+ zmul(t, t, a);
	+ \}
	+ zfree(a), zfree(one);
	+\}
	+\end{alltt}
	+\vspace{-1em}
	+
	+
	+
	\item \textbf{Modular tetration}

	Let \texttt{totient} be Euler's totient function.