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	Add exercise: [13] Modular generalised power towers - libzahl - big integer lib…
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	commit 18a0e23b2f2e37c56c4f0f9a3dd56c1c619a4468
	parent 7e8ea5ca2505686932b489e0aa1df0b3f0433e19
	Author: Mattias Andrée <[email protected]>
	Date: Sat, 30 Jul 2016 00:28:41 +0200

	Add exercise: [13] Modular generalised power towers

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

	Diffstat:
	M doc/exercises.tex \| 71 +++++++++++++++++++++++++++++…

	1 file changed, 70 insertions(+), 1 deletion(-)
	---
	diff --git a/doc/exercises.tex b/doc/exercises.tex
	@@ -290,6 +290,18 @@ You can assume $b > 0$ and $m > 0$. You can also assume
	unique pointers.


	+
	+\item {[\textit{13}]} \textbf{Modular generalised power towers}
	+
	+{\small\textit{This problem requires a working
	+solution for ``Modular tetration''.}}
	+
	+Modify your solution for ``Modular tetration'' to
	+evaluate any expression on the forms
	+$a^b,~a^{b^c},~a^{b^{c^d}},~\ldots \text{ mod } m$.
	+
	+
	+
	\end{enumerate}


	@@ -715,7 +727,7 @@ tetra(z_t r, z_t b, unsigned long n)
	\{
	zsetu(r, 1);
	while (n--)
	- pow(r, b, r);
	+ zpow(r, b, r);
	\}
	\end{alltt}
	\vspace{-1em}
	@@ -802,4 +814,61 @@ modtetra(z_t r, z_t b, unsigned long n, z_t m)



	+\item \textbf{Modular generalised power towers}
	+
	+Instead of the signature
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void modtetra(z_t r, z_t b, unsigned long n, z_t m);
	+\end{alltt}
	+\vspace{-1em}
	+
	+\noindent
	+you want to use the signature
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void modtower_(z_t r, z_t *a, size_t n, z_t m);
	+\end{alltt}
	+\vspace{-1em}
	+
	+Instead of using, \texttt{b} (in \texttt{modtetra}),
	+use \texttt{*a}. At every recursion, instead of
	+passing on \texttt{a}, pass on \texttt{a + 1}.
	+
	+The function \texttt{tetra} is modified into
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void tower(z_t r, z_t *a, size_t n)
	+ \{
	+ zsetu(r, 1);
	+ while (n--);
	+ zpow(r, a[n], r);
	+ \}
	+\end{alltt}
	+\vspace{-1em}
	+
	+\noindent
	+\texttt{cmp\_tetra} is changed analogously.
	+
	+To avoid problems in the evaluation, define
	+
	+\vspace{-1em}
	+\begin{alltt}
	+ void modtower(z_t r, z_t *a, size_t n, z_t m);
	+\end{alltt}
	+\vspace{-1em}
	+
	+\noindent
	+which cuts the power short at the first element
	+of of \texttt{a} that equals 1. For example, if
	+$a = \{2, 3, 4, 5, 1, 2, 3\}$, and $n = 7$
	+($n = \|a\|$), then \texttt{modtower}, sets $n = 4$,
	+and effectively $a = \{2, 3, 4, 5\}$. After this
	+\texttt{modtower} calls \texttt{modtower\_}.
	+
	+
	+
	\end{enumerate}