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284 lines
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284 lines
16 KiB
HTML
4 years ago
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 6.1.0.
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Copyright 1991, 1993-2015 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.3 or any later
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version published by the Free Software Foundation; with no Invariant Sections,
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with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover
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Texts being "You have freedom to copy and modify this GNU Manual, like GNU
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software". A copy of the license is included in
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GNU Free Documentation License. -->
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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Number Theoretic Functions (GNU MP 6.1.0)</title>
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<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
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<meta name="keywords" content="Number Theoretic Functions (GNU MP 6.1.0)">
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<meta name="resource-type" content="document">
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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<link href="index.html#Top" rel="start" title="Top">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="Integer-Functions.html#Integer-Functions" rel="up" title="Integer Functions">
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<link href="Integer-Comparisons.html#Integer-Comparisons" rel="next" title="Integer Comparisons">
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<link href="Integer-Roots.html#Integer-Roots" rel="prev" title="Integer Roots">
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</head>
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<body lang="en">
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<a name="Number-Theoretic-Functions"></a>
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<div class="header">
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<p>
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Next: <a href="Integer-Comparisons.html#Integer-Comparisons" accesskey="n" rel="next">Integer Comparisons</a>, Previous: <a href="Integer-Roots.html#Integer-Roots" accesskey="p" rel="prev">Integer Roots</a>, Up: <a href="Integer-Functions.html#Integer-Functions" accesskey="u" rel="up">Integer Functions</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<a name="Number-Theoretic-Functions-1"></a>
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<h3 class="section">5.9 Number Theoretic Functions</h3>
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<a name="index-Number-theoretic-functions"></a>
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<dl>
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<dt><a name="index-mpz_005fprobab_005fprime_005fp"></a>Function: <em>int</em> <strong>mpz_probab_prime_p</strong> <em>(const mpz_t <var>n</var>, int <var>reps</var>)</em></dt>
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<dd><a name="index-Prime-testing-functions"></a>
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<a name="index-Probable-prime-testing-functions"></a>
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<p>Determine whether <var>n</var> is prime. Return 2 if <var>n</var> is definitely prime,
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return 1 if <var>n</var> is probably prime (without being certain), or return 0 if
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<var>n</var> is definitely non-prime.
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</p>
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<p>This function performs some trial divisions, then <var>reps</var> Miller-Rabin
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probabilistic primality tests. A higher <var>reps</var> value will reduce the
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chances of a non-prime being identified as “probably prime”. A composite
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number will be identified as a prime with a probability of less than
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<em>4^(-<var>reps</var>)</em>. Reasonable values of <var>reps</var> are between 15
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and 50.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fnextprime"></a>Function: <em>void</em> <strong>mpz_nextprime</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op</var>)</em></dt>
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<dd><a name="index-Next-prime-function"></a>
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<p>Set <var>rop</var> to the next prime greater than <var>op</var>.
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</p>
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<p>This function uses a probabilistic algorithm to identify primes. For
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practical purposes it’s adequate, the chance of a composite passing will be
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extremely small.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fgcd"></a>Function: <em>void</em> <strong>mpz_gcd</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op1</var>, const mpz_t <var>op2</var>)</em></dt>
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<dd><a name="index-Greatest-common-divisor-functions"></a>
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<a name="index-GCD-functions"></a>
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<p>Set <var>rop</var> to the greatest common divisor of <var>op1</var> and <var>op2</var>. The
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result is always positive even if one or both input operands are negative.
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Except if both inputs are zero; then this function defines <em>gcd(0,0) = 0</em>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fgcd_005fui"></a>Function: <em>unsigned long int</em> <strong>mpz_gcd_ui</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op1</var>, unsigned long int <var>op2</var>)</em></dt>
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<dd><p>Compute the greatest common divisor of <var>op1</var> and <var>op2</var>. If
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<var>rop</var> is not <code>NULL</code>, store the result there.
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</p>
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<p>If the result is small enough to fit in an <code>unsigned long int</code>, it is
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returned. If the result does not fit, 0 is returned, and the result is equal
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to the argument <var>op1</var>. Note that the result will always fit if <var>op2</var>
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is non-zero.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fgcdext"></a>Function: <em>void</em> <strong>mpz_gcdext</strong> <em>(mpz_t <var>g</var>, mpz_t <var>s</var>, mpz_t <var>t</var>, const mpz_t <var>a</var>, const mpz_t <var>b</var>)</em></dt>
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<dd><a name="index-Extended-GCD"></a>
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<a name="index-GCD-extended"></a>
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<p>Set <var>g</var> to the greatest common divisor of <var>a</var> and <var>b</var>, and in
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addition set <var>s</var> and <var>t</var> to coefficients satisfying
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<em><var>a</var>*<var>s</var> + <var>b</var>*<var>t</var> = <var>g</var></em>.
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The value in <var>g</var> is always positive, even if one or both of <var>a</var> and
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<var>b</var> are negative (or zero if both inputs are zero). The values in <var>s</var>
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and <var>t</var> are chosen such that normally, <em>abs(<var>s</var>) <
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abs(<var>b</var>) / (2 <var>g</var>)</em> and <em>abs(<var>t</var>) < abs(<var>a</var>)
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/ (2 <var>g</var>)</em>, and these relations define <var>s</var> and <var>t</var> uniquely. There
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are a few exceptional cases:
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</p>
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<p>If <em>abs(<var>a</var>) = abs(<var>b</var>)</em>, then <em><var>s</var> = 0</em>,
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<em><var>t</var> = sgn(<var>b</var>)</em>.
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</p>
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<p>Otherwise, <em><var>s</var> = sgn(<var>a</var>)</em> if <em><var>b</var> = 0</em> or
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<em>abs(<var>b</var>) = 2 <var>g</var></em>, and <em><var>t</var> = sgn(<var>b</var>)</em> if
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<em><var>a</var> = 0</em> or <em>abs(<var>a</var>) = 2 <var>g</var></em>.
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</p>
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<p>In all cases, <em><var>s</var> = 0</em> if and only if <em><var>g</var> =
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abs(<var>b</var>)</em>, i.e., if <var>b</var> divides <var>a</var> or <em><var>a</var> = <var>b</var>
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= 0</em>.
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</p>
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<p>If <var>t</var> is <code>NULL</code> then that value is not computed.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005flcm"></a>Function: <em>void</em> <strong>mpz_lcm</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op1</var>, const mpz_t <var>op2</var>)</em></dt>
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<dt><a name="index-mpz_005flcm_005fui"></a>Function: <em>void</em> <strong>mpz_lcm_ui</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op1</var>, unsigned long <var>op2</var>)</em></dt>
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<dd><a name="index-Least-common-multiple-functions"></a>
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<a name="index-LCM-functions"></a>
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<p>Set <var>rop</var> to the least common multiple of <var>op1</var> and <var>op2</var>.
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<var>rop</var> is always positive, irrespective of the signs of <var>op1</var> and
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<var>op2</var>. <var>rop</var> will be zero if either <var>op1</var> or <var>op2</var> is zero.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005finvert"></a>Function: <em>int</em> <strong>mpz_invert</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op1</var>, const mpz_t <var>op2</var>)</em></dt>
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<dd><a name="index-Modular-inverse-functions"></a>
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<a name="index-Inverse-modulo-functions"></a>
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<p>Compute the inverse of <var>op1</var> modulo <var>op2</var> and put the result in
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<var>rop</var>. If the inverse exists, the return value is non-zero and <var>rop</var>
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will satisfy <em>0 <= <var>rop</var> < abs(<var>op2</var>)</em> (with <em><var>rop</var>
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= 0</em> possible only when <em>abs(<var>op2</var>) = 1</em>, i.e., in the
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somewhat degenerate zero ring). If an inverse doesn’t
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exist the return value is zero and <var>rop</var> is undefined. The behaviour of
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this function is undefined when <var>op2</var> is zero.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fjacobi"></a>Function: <em>int</em> <strong>mpz_jacobi</strong> <em>(const mpz_t <var>a</var>, const mpz_t <var>b</var>)</em></dt>
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<dd><a name="index-Jacobi-symbol-functions"></a>
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<p>Calculate the Jacobi symbol <em>(<var>a</var>/<var>b</var>)</em>. This is defined only for <var>b</var> odd.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005flegendre"></a>Function: <em>int</em> <strong>mpz_legendre</strong> <em>(const mpz_t <var>a</var>, const mpz_t <var>p</var>)</em></dt>
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<dd><a name="index-Legendre-symbol-functions"></a>
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<p>Calculate the Legendre symbol <em>(<var>a</var>/<var>p</var>)</em>. This is defined only for <var>p</var> an odd positive
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prime, and for such <var>p</var> it’s identical to the Jacobi symbol.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fkronecker"></a>Function: <em>int</em> <strong>mpz_kronecker</strong> <em>(const mpz_t <var>a</var>, const mpz_t <var>b</var>)</em></dt>
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<dt><a name="index-mpz_005fkronecker_005fsi"></a>Function: <em>int</em> <strong>mpz_kronecker_si</strong> <em>(const mpz_t <var>a</var>, long <var>b</var>)</em></dt>
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<dt><a name="index-mpz_005fkronecker_005fui"></a>Function: <em>int</em> <strong>mpz_kronecker_ui</strong> <em>(const mpz_t <var>a</var>, unsigned long <var>b</var>)</em></dt>
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<dt><a name="index-mpz_005fsi_005fkronecker"></a>Function: <em>int</em> <strong>mpz_si_kronecker</strong> <em>(long <var>a</var>, const mpz_t <var>b</var>)</em></dt>
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<dt><a name="index-mpz_005fui_005fkronecker"></a>Function: <em>int</em> <strong>mpz_ui_kronecker</strong> <em>(unsigned long <var>a</var>, const mpz_t <var>b</var>)</em></dt>
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<dd><a name="index-Kronecker-symbol-functions"></a>
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<p>Calculate the Jacobi symbol <em>(<var>a</var>/<var>b</var>)</em> with the Kronecker extension <em>(a/2)=(2/a)</em> when <em>a</em> odd, or
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<em>(a/2)=0</em> when <em>a</em> even.
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</p>
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<p>When <var>b</var> is odd the Jacobi symbol and Kronecker symbol are
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identical, so <code>mpz_kronecker_ui</code> etc can be used for mixed
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precision Jacobi symbols too.
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</p>
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<p>For more information see Henri Cohen section 1.4.2 (see <a href="References.html#References">References</a>),
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or any number theory textbook. See also the example program
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<samp>demos/qcn.c</samp> which uses <code>mpz_kronecker_ui</code>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fremove"></a>Function: <em>mp_bitcnt_t</em> <strong>mpz_remove</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>op</var>, const mpz_t <var>f</var>)</em></dt>
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<dd><a name="index-Remove-factor-functions"></a>
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<a name="index-Factor-removal-functions"></a>
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<p>Remove all occurrences of the factor <var>f</var> from <var>op</var> and store the
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result in <var>rop</var>. The return value is how many such occurrences were
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removed.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005ffac_005fui"></a>Function: <em>void</em> <strong>mpz_fac_ui</strong> <em>(mpz_t <var>rop</var>, unsigned long int <var>n</var>)</em></dt>
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<dt><a name="index-mpz_005f2fac_005fui"></a>Function: <em>void</em> <strong>mpz_2fac_ui</strong> <em>(mpz_t <var>rop</var>, unsigned long int <var>n</var>)</em></dt>
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<dt><a name="index-mpz_005fmfac_005fuiui"></a>Function: <em>void</em> <strong>mpz_mfac_uiui</strong> <em>(mpz_t <var>rop</var>, unsigned long int <var>n</var>, unsigned long int <var>m</var>)</em></dt>
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<dd><a name="index-Factorial-functions"></a>
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<p>Set <var>rop</var> to the factorial of <var>n</var>: <code>mpz_fac_ui</code> computes the plain factorial <var>n</var>!,
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<code>mpz_2fac_ui</code> computes the double-factorial <var>n</var>!!, and <code>mpz_mfac_uiui</code> the
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<var>m</var>-multi-factorial <em><var>n</var>!^(<var>m</var>)</em>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fprimorial_005fui"></a>Function: <em>void</em> <strong>mpz_primorial_ui</strong> <em>(mpz_t <var>rop</var>, unsigned long int <var>n</var>)</em></dt>
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<dd><a name="index-Primorial-functions"></a>
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<p>Set <var>rop</var> to the primorial of <var>n</var>, i.e. the product of all positive
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prime numbers <em><=<var>n</var></em>.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005fbin_005fui"></a>Function: <em>void</em> <strong>mpz_bin_ui</strong> <em>(mpz_t <var>rop</var>, const mpz_t <var>n</var>, unsigned long int <var>k</var>)</em></dt>
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<dt><a name="index-mpz_005fbin_005fuiui"></a>Function: <em>void</em> <strong>mpz_bin_uiui</strong> <em>(mpz_t <var>rop</var>, unsigned long int <var>n</var>, unsigned long int <var>k</var><!-- /@w -->)</em></dt>
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<dd><a name="index-Binomial-coefficient-functions"></a>
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<p>Compute the binomial coefficient <em><var>n</var> over
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<var>k</var></em> and store the result in <var>rop</var>. Negative values of <var>n</var> are
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supported by <code>mpz_bin_ui</code>, using the identity
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<em>bin(-n,k) = (-1)^k * bin(n+k-1,k)</em>, see Knuth volume 1 section 1.2.6
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part G.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005ffib_005fui"></a>Function: <em>void</em> <strong>mpz_fib_ui</strong> <em>(mpz_t <var>fn</var>, unsigned long int <var>n</var>)</em></dt>
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<dt><a name="index-mpz_005ffib2_005fui"></a>Function: <em>void</em> <strong>mpz_fib2_ui</strong> <em>(mpz_t <var>fn</var>, mpz_t <var>fnsub1</var>, unsigned long int <var>n</var>)</em></dt>
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<dd><a name="index-Fibonacci-sequence-functions"></a>
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<p><code>mpz_fib_ui</code> sets <var>fn</var> to to <em>F[n]</em>, the <var>n</var>’th Fibonacci
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number. <code>mpz_fib2_ui</code> sets <var>fn</var> to <em>F[n]</em>, and <var>fnsub1</var> to
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<em>F[n-1]</em>.
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</p>
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<p>These functions are designed for calculating isolated Fibonacci numbers. When
|
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a sequence of values is wanted it’s best to start with <code>mpz_fib2_ui</code> and
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iterate the defining <em>F[n+1]=F[n]+F[n-1]</em> or
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similar.
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</p></dd></dl>
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<dl>
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<dt><a name="index-mpz_005flucnum_005fui"></a>Function: <em>void</em> <strong>mpz_lucnum_ui</strong> <em>(mpz_t <var>ln</var>, unsigned long int <var>n</var>)</em></dt>
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<dt><a name="index-mpz_005flucnum2_005fui"></a>Function: <em>void</em> <strong>mpz_lucnum2_ui</strong> <em>(mpz_t <var>ln</var>, mpz_t <var>lnsub1</var>, unsigned long int <var>n</var>)</em></dt>
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<dd><a name="index-Lucas-number-functions"></a>
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<p><code>mpz_lucnum_ui</code> sets <var>ln</var> to to <em>L[n]</em>, the <var>n</var>’th Lucas
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number. <code>mpz_lucnum2_ui</code> sets <var>ln</var> to <em>L[n]</em>, and <var>lnsub1</var>
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to <em>L[n-1]</em>.
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||
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</p>
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<p>These functions are designed for calculating isolated Lucas numbers. When a
|
||
|
|
sequence of values is wanted it’s best to start with <code>mpz_lucnum2_ui</code> and
|
||
|
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iterate the defining <em>L[n+1]=L[n]+L[n-1]</em> or
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similar.
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||
|
|
</p>
|
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<p>The Fibonacci numbers and Lucas numbers are related sequences, so it’s never
|
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necessary to call both <code>mpz_fib2_ui</code> and <code>mpz_lucnum2_ui</code>. The
|
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formulas for going from Fibonacci to Lucas can be found in <a href="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm">Lucas Numbers Algorithm</a>, the reverse is straightforward too.
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||
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</p></dd></dl>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Integer-Comparisons.html#Integer-Comparisons" accesskey="n" rel="next">Integer Comparisons</a>, Previous: <a href="Integer-Roots.html#Integer-Roots" accesskey="p" rel="prev">Integer Roots</a>, Up: <a href="Integer-Functions.html#Integer-Functions" accesskey="u" rel="up">Integer Functions</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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