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	Add exercise: [20] Fast primality test with bounded perfection - libzahl - big …
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	commit 076e4e3284039e1229bc7f99232e415cdc44711d
	parent 611db6bdb6a5a08b628f571451a94a1147a0e16f
	Author: Mattias Andrée <[email protected]>
	Date: Mon, 25 Jul 2016 01:13:00 +0200

	Add exercise: [20] Fast primality test with bounded perfection

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

	Diffstat:
	M doc/exercises.tex \| 36 +++++++++++++++++++++++++++++…

	1 file changed, 36 insertions(+), 0 deletions(-)
	---
	diff --git a/doc/exercises.tex b/doc/exercises.tex
	@@ -180,6 +180,14 @@ is not part of the difficulty rating of this problem.)



	+\item {[\textit{20}]} \textbf{Fast primality test with bounded perfection}
	+
	+Implement a primality test that is both very fast and
	+never returns \texttt{PROBABLY\_PRIME} for input less
	+than or equal to a preselected number.
	+
	+
	+
	\end{enumerate}


	@@ -433,4 +441,32 @@ Mersenne number) to first check that $n$ is prime.



	+\item \textbf{Fast primality test with bounded perfection}
	+
	+First we select a fast primality test. We can use
	+$2^p \equiv 2 ~(\texttt{Mod}~ p) ~\forall~ p \in \textbf{P}$,
	+as describe in the solution for the problem
	+\textit{Fast primality test}.
	+
	+Next, we use this to generate a large list of primes and
	+pseudoprimes. Use a perfect primality test, such as a
	+naïve test or the AKS primality test, to filter out all
	+primes and retain only the pseudoprimes. This is not in
	+runtime so it does not matter that this is slow, but to
	+speed it up, we can use a probabilistic test such the
	+Miller–Rabin primality test (\texttt{zptest}) before we
	+use the perfect test.
	+
	+Now that we have a quite large — but not humongous — list
	+of pseudoprimes, we can incorporate it into our fast
	+primality test. For any input that passes the test, and
	+is less or equal to the largest pseudoprime we found,
	+binary search our list of pseudoprime for the input.
	+
	+For input, larger than our limit, that passes the test,
	+we can run it through \texttt{zptest} to reduce the
	+number of false positives.
	+
	+
	+
	\end{enumerate}