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Next: <a href="Factorial-Algorithm.html#Factorial-Algorithm" accesskey="n" rel="next">Factorial Algorithm</a>, Previous: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="p" rel="prev">Other Algorithms</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Prime-Testing"></a>
<h4 class="subsection">15.7.1 Prime Testing</h4>
<a name="index-Prime-testing-algorithms"></a>
<p>The primality testing in <code>mpz_probab_prime_p</code> (see <a href="Number-Theoretic-Functions.html#Number-Theoretic-Functions">Number Theoretic Functions</a>) first does some trial division by small factors and then uses the
Miller-Rabin probabilistic primality testing algorithm, as described in Knuth
section 4.5.4 algorithm P (see <a href="References.html#References">References</a>).
</p>
<p>For an odd input <em>n</em>, and with <em>n = q*2^k+1</em> where
<em>q</em> is odd, this algorithm selects a random base <em>x</em> and tests
whether <em>x^q mod n</em> is 1 or <em>-1</em>, or an <em>x^(q*2^j) mod n</em> is <em>1</em>, for <em>1<=j<=k</em>. If so then <em>n</em>
is probably prime, if not then <em>n</em> is definitely composite.
</p>
<p>Any prime <em>n</em> will pass the test, but some composites do too. Such
composites are known as strong pseudoprimes to base <em>x</em>. No <em>n</em> is
a strong pseudoprime to more than <em>1/4</em> of all bases (see Knuth exercise
22), hence with <em>x</em> chosen at random there’s no more than a <em>1/4</em>
chance a “probable prime” will in fact be composite.
</p>
<p>In fact strong pseudoprimes are quite rare, making the test much more
powerful than this analysis would suggest, but <em>1/4</em> is all that’s proven
for an arbitrary <em>n</em>.
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