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<p><a id="X853DF11B80068ED5" name="X853DF11B80068ED5"></a></p>
<div class="ChapSects"><a href="chap14.html#X853DF11B80068ED5">14 <span class="Heading">Integers</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X838230CE810107A3">14.1 <span class="Heading">Integers: Global Variables</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7E20D82B79DE5129">14.1-1 Integers</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X818683B17F8C97F3">14.1-2 IsIntegers</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X80CF510B8080C7CA">14.2 <span class="Heading">Elementary Operations for Integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X87AEADF07DC8303B">14.2-1 IsInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X82A854757DFA9C76">14.2-2 IsPosInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X87CA734380B5F68C">14.2-3 Int</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X87DD1EEE7EF18036">14.2-4 IsEvenInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X8621BA927CD12EFB">14.2-5 IsOddInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X782095927FB9F1DB">14.2-6 AbsInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X842614817FE48D62">14.2-7 SignInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X8197C4E882BAF14E">14.2-8 LogInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X83D9B5C87EEA2A77">14.2-9 RootInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7F98A0CE7B9FD366">14.2-10 SmallestRootInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X862D1BD786EFFDA9">14.2-11 ListOfDigits</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X8185784B7E228DEA">14.2-12 Random</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7A9FD25D81D88D1B">14.3 <span class="Heading">Quotients and Remainders</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X849D0F807F697D35">14.3-1 QuoInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X795170A385AC8FEE">14.3-2 BestQuoInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X805ADD5A826D844D">14.3-3 RemInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7A4FEFCA8128E3C3">14.3-4 GcdInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X8775930486BD0C5B">14.3-5 Gcdex</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7B33143E78A8DDE3">14.3-6 LcmInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X79B466E984CD52D4">14.3-7 CoefficientsQadic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X83124F86839DC7E6">14.3-8 CoefficientsMultiadic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X84A1900E82902B5F">14.3-9 ChineseRem</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7E404B1183DBC82A">14.3-10 PowerModInt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X82005E587F0CB02A">14.4 <span class="Heading">Prime Integers and Factorization</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X86F5E4CD82FEB9F4">14.4-1 Primes</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X78FDA4437EDCA70C">14.4-2 IsPrimeInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7CD977B17B4A7A4B">14.4-3 PrimalityProof</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X8443125D7FD6F2A6">14.4-4 IsPrimePowerInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X78744C367A94C69F">14.4-5 NextPrimeInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X819060E17E83728A">14.4-6 PrevPrimeInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X82C989DB84744B36">14.4-7 FactorsInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X80E7A5D381C64CC9">14.4-8 PrimeDivisors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X786FF92C7C54BF97">14.4-9 PartialFactorization</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X803D431087B6FF28">14.4-10 PrintFactorsInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X82148B347E294C87">14.4-11 PrimePowersInt</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X809E0E1B83AF7695">14.4-12 DivisorsInt</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X864BF040862409FC">14.5 <span class="Heading">Residue Class Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X87B1210B8581D5B2"><code>14.5-1 \mod</code></a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X79CE76AD82B3E2B2">14.5-2 ZmodnZ</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X838F36507D985EDA">14.5-3 ZmodnZObj</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7D0107DD79753901">14.5-4 IsZmodnZObj</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7904B6D681EBF091">14.6 <span class="Heading">Check Digits</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X82BABA8F868BD425">14.6-1 CheckDigitISBN</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X85F1A6A5870485B9">14.6-2 CheckDigitTestFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X85361FAE8088C006">14.7 <span class="Heading">Random Sources</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X82E31A697E389F1D">14.7-1 IsRandomSource</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X821004F286282D49">14.7-2 Random</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X819E3E3080297347">14.7-3 State</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7F772E2686B35865">14.7-4 IsMersenneTwister</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X7CB0B5BC82F8FD8F">14.7-5 RandomSource</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7A0311DF78DB4FD8">14.8 <span class="Heading">Bitfields</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X85C7BD9E7FCC6C10">14.8-1 MakeBitfields</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X8068CE3781F4003C">14.8-2 BuildBitfields</a></span>
</div></div>
</div>
<h3>14 <span class="Heading">Integers</span></h3>
<p>One of the most fundamental datatypes in every programming language is the integer type. <strong class="pkg">GAP</strong> is no exception.</p>
<p><strong class="pkg">GAP</strong> integers are entered as a sequence of decimal digits optionally preceded by a "<code class="code">+</code>" sign for positive integers or a "<code class="code">-</code>" sign for negative integers. The size of integers in <strong class="pkg">GAP</strong> is only limited by the amount of available memory, so you can compute with integers having thousands of digits.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">-1234;</span>
-1234
<span class="GAPprompt">gap></span> <span class="GAPinput">123456789012345678901234567890123456789012345678901234567890;</span>
123456789012345678901234567890123456789012345678901234567890
</pre></div>
<p>Many more functions that are mainly related to the prime residue group of integers modulo an integer are described in chapter <a href="chap15.html#X7FB995737B7ED8A2"><span class="RefLink">15</span></a>, and functions dealing with combinatorics can be found in chapter <a href="chap16.html#X7BDA99EE7CEADA7C"><span class="RefLink">16</span></a>.</p>
<p><a id="X838230CE810107A3" name="X838230CE810107A3"></a></p>
<h4>14.1 <span class="Heading">Integers: Global Variables</span></h4>
<p><a id="X7E20D82B79DE5129" name="X7E20D82B79DE5129"></a></p>
<h5>14.1-1 Integers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Integers</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PositiveIntegers</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NonnegativeIntegers</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>These global variables represent the ring of integers and the semirings of positive and nonnegative integers, respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( Integers ); 2 in Integers;</span>
infinity
true
</pre></div>
<p><code class="func">Integers</code> is a subset of <code class="func">Rationals</code> (<a href="chap17.html#X7B6029D18570C08A"><span class="RefLink">17.1-1</span></a>), which is a subset of <code class="func">Cyclotomics</code> (<a href="chap18.html#X863D1E017BC9EB7F"><span class="RefLink">18.1-2</span></a>). See Chapter <a href="chap18.html#X7DFC03C187DE4841"><span class="RefLink">18</span></a> for arithmetic operations and comparison of integers.</p>
<p><a id="X818683B17F8C97F3" name="X818683B17F8C97F3"></a></p>
<h5>14.1-2 IsIntegers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIntegers</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPositiveIntegers</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNonnegativeIntegers</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>are the defining categories for the domains <code class="func">Integers</code> (<a href="chap14.html#X7E20D82B79DE5129"><span class="RefLink">14.1-1</span></a>), <code class="func">PositiveIntegers</code> (<a href="chap14.html#X7E20D82B79DE5129"><span class="RefLink">14.1-1</span></a>), and <code class="func">NonnegativeIntegers</code> (<a href="chap14.html#X7E20D82B79DE5129"><span class="RefLink">14.1-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIntegers( Integers ); IsIntegers( Rationals ); IsIntegers( 7 );</span>
true
false
false
</pre></div>
<p><a id="X80CF510B8080C7CA" name="X80CF510B8080C7CA"></a></p>
<h4>14.2 <span class="Heading">Elementary Operations for Integers</span></h4>
<p><a id="X87AEADF07DC8303B" name="X87AEADF07DC8303B"></a></p>
<h5>14.2-1 IsInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsInt</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Every rational integer lies in the category <code class="func">IsInt</code>, which is a subcategory of <code class="func">IsRat</code> (<a href="chap17.html#X7ED018F5794935F7"><span class="RefLink">17.2-1</span></a>).</p>
<p><a id="X82A854757DFA9C76" name="X82A854757DFA9C76"></a></p>
<h5>14.2-2 IsPosInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPosInt</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Every positive integer lies in the category <code class="func">IsPosInt</code>.</p>
<p><a id="X87CA734380B5F68C" name="X87CA734380B5F68C"></a></p>
<h5>14.2-3 Int</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Int</code>( <var class="Arg">elm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">Int</code> returns an integer <code class="code">int</code> whose meaning depends on the type of <var class="Arg">elm</var>. For example:</p>
<p>If <var class="Arg">elm</var> is a rational number (see Chapter <a href="chap17.html#X87003045878E74DF"><span class="RefLink">17</span></a>) then <code class="code">int</code> is the integer part of the quotient of numerator and denominator of <var class="Arg">elm</var> (see <code class="func">QuoInt</code> (<a href="chap14.html#X849D0F807F697D35"><span class="RefLink">14.3-1</span></a>)).</p>
<p>If <var class="Arg">elm</var> is an element of a finite prime field (see Chapter <a href="chap59.html#X7893ABF67A028802"><span class="RefLink">59</span></a>) then <code class="code">int</code> is the smallest nonnegative integer such that <code class="code"><var class="Arg">elm</var> = int * One( <var class="Arg">elm</var> )</code>.</p>
<p>If <var class="Arg">elm</var> is a string (see Chapter <a href="chap27.html#X7D28329B7EDB8F47"><span class="RefLink">27</span></a>) consisting entirely of decimal digits <code class="code">'0'</code>, <code class="code">'1'</code>, <span class="SimpleMath">...</span>, <code class="code">'9'</code>, and optionally a sign <code class="code">'-'</code> (at the first position), then <code class="code">int</code> is the integer described by this string. For all other strings, <code class="code">fail</code> is returned. See <code class="func">Int</code> (<a href="chap27.html#X7B6D118184F692A0"><span class="RefLink">27.9-1</span></a>).</p>
<p>The operation <code class="func">String</code> (<a href="chap27.html#X81FB5BE27903EC32"><span class="RefLink">27.7-6</span></a>) can be used to compute a string for rational integers, in fact for all cyclotomics.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Int( 4/3 ); Int( -2/3 );</span>
1
0
<span class="GAPprompt">gap></span> <span class="GAPinput">int:= Int( Z(5) ); int * One( Z(5) );</span>
2
Z(5)
<span class="GAPprompt">gap></span> <span class="GAPinput">Int( "12345" ); Int( "-27" ); Int( "-27/3" );</span>
12345
-27
fail
</pre></div>
<p><a id="X87DD1EEE7EF18036" name="X87DD1EEE7EF18036"></a></p>
<h5>14.2-4 IsEvenInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEvenInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests if the integer <var class="Arg">n</var> is divisible by 2.</p>
<p><a id="X8621BA927CD12EFB" name="X8621BA927CD12EFB"></a></p>
<h5>14.2-5 IsOddInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOddInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests if the integer <var class="Arg">n</var> is not divisible by 2.</p>
<p><a id="X782095927FB9F1DB" name="X782095927FB9F1DB"></a></p>
<h5>14.2-6 AbsInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbsInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">AbsInt</code> returns the absolute value of the integer <var class="Arg">n</var>, i.e., <var class="Arg">n</var> if <var class="Arg">n</var> is positive, -<var class="Arg">n</var> if <var class="Arg">n</var> is negative and 0 if <var class="Arg">n</var> is 0.</p>
<p><code class="func">AbsInt</code> is a special case of the general operation <code class="func">EuclideanDegree</code> (<a href="chap56.html#X784234088350D4E4"><span class="RefLink">56.6-2</span></a>).</p>
<p>See also <code class="func">AbsoluteValue</code> (<a href="chap18.html#X81DD58BB81FB3426"><span class="RefLink">18.1-8</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbsInt( 33 );</span>
33
<span class="GAPprompt">gap></span> <span class="GAPinput">AbsInt( -214378 );</span>
214378
<span class="GAPprompt">gap></span> <span class="GAPinput">AbsInt( 0 );</span>
0
</pre></div>
<p><a id="X842614817FE48D62" name="X842614817FE48D62"></a></p>
<h5>14.2-7 SignInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SignInt</code> returns the sign of the integer <var class="Arg">n</var>, i.e., 1 if <var class="Arg">n</var> is positive, -1 if <var class="Arg">n</var> is negative and 0 if <var class="Arg">n</var> is 0.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SignInt( 33 );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">SignInt( -214378 );</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">SignInt( 0 );</span>
0
</pre></div>
<p><a id="X8197C4E882BAF14E" name="X8197C4E882BAF14E"></a></p>
<h5>14.2-8 LogInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LogInt</code>( <var class="Arg">n</var>, <var class="Arg">base</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">LogInt</code> returns the integer part of the logarithm of the positive integer <var class="Arg">n</var> with respect to the positive integer <var class="Arg">base</var>, i.e., the largest positive integer <span class="SimpleMath">e</span> such that <span class="SimpleMath"><var class="Arg">base</var>^e ≤ <var class="Arg">n</var></span>. The function <code class="func">LogInt</code> will signal an error if either <var class="Arg">n</var> or <var class="Arg">base</var> is not positive.</p>
<p>For <var class="Arg">base</var> <span class="SimpleMath">= 2</span> this is very efficient because the internal binary representation of the integer is used.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LogInt( 1030, 2 );</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">2^10;</span>
1024
<span class="GAPprompt">gap></span> <span class="GAPinput">LogInt( 1, 10 );</span>
0
</pre></div>
<p><a id="X83D9B5C87EEA2A77" name="X83D9B5C87EEA2A77"></a></p>
<h5>14.2-9 RootInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RootInt</code>( <var class="Arg">n</var>[, <var class="Arg">k</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">RootInt</code> returns the integer part of the <var class="Arg">k</var>th root of the integer <var class="Arg">n</var>. If the optional integer argument <var class="Arg">k</var> is not given it defaults to 2, i.e., <code class="func">RootInt</code> returns the integer part of the square root in this case.</p>
<p>If <var class="Arg">n</var> is positive, <code class="func">RootInt</code> returns the largest positive integer <span class="SimpleMath">r</span> such that <span class="SimpleMath">r^<var class="Arg">k</var> ≤ <var class="Arg">n</var></span>. If <var class="Arg">n</var> is negative and <var class="Arg">k</var> is odd <code class="func">RootInt</code> returns <code class="code">-RootInt( -<var class="Arg">n</var>, <var class="Arg">k</var> )</code>. If <var class="Arg">n</var> is negative and <var class="Arg">k</var> is even <code class="func">RootInt</code> will cause an error. <code class="func">RootInt</code> will also cause an error if <var class="Arg">k</var> is 0 or negative.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RootInt( 361 );</span>
19
<span class="GAPprompt">gap></span> <span class="GAPinput">RootInt( 2 * 10^12 );</span>
1414213
<span class="GAPprompt">gap></span> <span class="GAPinput">RootInt( 17000, 5 );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">7^5;</span>
16807
</pre></div>
<p><a id="X7F98A0CE7B9FD366" name="X7F98A0CE7B9FD366"></a></p>
<h5>14.2-10 SmallestRootInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestRootInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">SmallestRootInt</code> returns the smallest root of the integer <var class="Arg">n</var>.</p>
<p>The smallest root of an integer <var class="Arg">n</var> is the integer <span class="SimpleMath">r</span> of smallest absolute value for which a positive integer <span class="SimpleMath">k</span> exists such that <span class="SimpleMath"><var class="Arg">n</var> = r^k</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestRootInt( 2^30 );</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestRootInt( -(2^30) );</span>
-4
</pre></div>
<p>Note that <span class="SimpleMath">(-2)^30 = +(2^30)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestRootInt( 279936 );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">LogInt( 279936, 6 );</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestRootInt( 1001 );</span>
1001
</pre></div>
<p><a id="X862D1BD786EFFDA9" name="X862D1BD786EFFDA9"></a></p>
<h5>14.2-11 ListOfDigits</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ListOfDigits</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a positive integer <var class="Arg">n</var> this function returns a list <var class="Arg">l</var>, consisting of the digits of <var class="Arg">n</var> in decimal representation.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ListOfDigits(3142); </span>
[ 3, 1, 4, 2 ]
</pre></div>
<p><a id="X8185784B7E228DEA" name="X8185784B7E228DEA"></a></p>
<h5>14.2-12 Random</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">Integers</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><code class="func">Random</code> for integers returns pseudo random integers between <span class="SimpleMath">-10</span> and <span class="SimpleMath">10</span> distributed according to a binomial distribution. To generate uniformly distributed integers from a range, use the construction <code class="code">Random( [ <var class="Arg">low</var> .. <var class="Arg">high</var> ] )</code> (see <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>)).</p>
<p><a id="X7A9FD25D81D88D1B" name="X7A9FD25D81D88D1B"></a></p>
<h4>14.3 <span class="Heading">Quotients and Remainders</span></h4>
<p><a id="X849D0F807F697D35" name="X849D0F807F697D35"></a></p>
<h5>14.3-1 QuoInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuoInt</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">QuoInt</code> returns the integer part of the quotient of its integer operands.</p>
<p>If <var class="Arg">n</var> and <var class="Arg">m</var> are positive, <code class="func">QuoInt</code> returns the largest positive integer <span class="SimpleMath">q</span> such that <span class="SimpleMath">q * <var class="Arg">m</var> ≤ <var class="Arg">n</var></span>. If <var class="Arg">n</var> or <var class="Arg">m</var> or both are negative the absolute value of the integer part of the quotient is the quotient of the absolute values of <var class="Arg">n</var> and <var class="Arg">m</var>, and the sign of it is the product of the signs of <var class="Arg">n</var> and <var class="Arg">m</var>.</p>
<p><code class="func">QuoInt</code> is used in a method for the general operation <code class="func">EuclideanQuotient</code> (<a href="chap56.html#X7A93FA788318B147"><span class="RefLink">56.6-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QuoInt(5,3); QuoInt(-5,3); QuoInt(5,-3); QuoInt(-5,-3);</span>
1
-1
-1
1
</pre></div>
<p><a id="X795170A385AC8FEE" name="X795170A385AC8FEE"></a></p>
<h5>14.3-2 BestQuoInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BestQuoInt</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">BestQuoInt</code> returns the best quotient <span class="SimpleMath">q</span> of the integers <var class="Arg">n</var> and <var class="Arg">m</var>. This is the quotient such that <span class="SimpleMath"><var class="Arg">n</var>-q*<var class="Arg">m</var></span> has minimal absolute value. If there are two quotients whose remainders have the same absolute value, then the quotient with the smaller absolute value is chosen.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BestQuoInt( 5, 3 ); BestQuoInt( -5, 3 );</span>
2
-2
</pre></div>
<p><a id="X805ADD5A826D844D" name="X805ADD5A826D844D"></a></p>
<h5>14.3-3 RemInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RemInt</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">RemInt</code> returns the remainder of its two integer operands.</p>
<p>If <var class="Arg">m</var> is not equal to zero, <code class="func">RemInt</code> returns <code class="code"><var class="Arg">n</var> - <var class="Arg">m</var> * QuoInt( <var class="Arg">n</var>, <var class="Arg">m</var> )</code>. Note that the rules given for <code class="func">QuoInt</code> (<a href="chap14.html#X849D0F807F697D35"><span class="RefLink">14.3-1</span></a>) imply that the return value of <code class="func">RemInt</code> has the same sign as <var class="Arg">n</var> and its absolute value is strictly less than the absolute value of <var class="Arg">m</var>. Note also that the return value equals <code class="code"><var class="Arg">n</var> mod <var class="Arg">m</var></code> when both <var class="Arg">n</var> and <var class="Arg">m</var> are nonnegative. Dividing by <code class="code">0</code> signals an error.</p>
<p><code class="func">RemInt</code> is used in a method for the general operation <code class="func">EuclideanRemainder</code> (<a href="chap56.html#X7B5E9639865E91BA"><span class="RefLink">56.6-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RemInt(5,3); RemInt(-5,3); RemInt(5,-3); RemInt(-5,-3);</span>
2
-2
2
-2
</pre></div>
<p><a id="X7A4FEFCA8128E3C3" name="X7A4FEFCA8128E3C3"></a></p>
<h5>14.3-4 GcdInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GcdInt</code>( <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">GcdInt</code> returns the greatest common divisor of its two integer operands <var class="Arg">m</var> and <var class="Arg">n</var>, i.e., the greatest integer that divides both <var class="Arg">m</var> and <var class="Arg">n</var>. The greatest common divisor is never negative, even if the arguments are. We define <code class="code">GcdInt( <var class="Arg">m</var>, 0 ) = GcdInt( 0, <var class="Arg">m</var> ) = AbsInt( <var class="Arg">m</var> )</code> and <code class="code">GcdInt( 0, 0 ) = 0</code>.</p>
<p><code class="func">GcdInt</code> is a method used by the general function <code class="func">Gcd</code> (<a href="chap56.html#X7DE207718456F98F"><span class="RefLink">56.7-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GcdInt( 123, 66 );</span>
3
</pre></div>
<p><a id="X8775930486BD0C5B" name="X8775930486BD0C5B"></a></p>
<h5>14.3-5 Gcdex</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Gcdex</code>( <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a record <code class="code">g</code> describing the extended greatest common divisor of <var class="Arg">m</var> and <var class="Arg">n</var>. The component <code class="code">gcd</code> is this gcd, the components <code class="code">coeff1</code> and <code class="code">coeff2</code> are integer cofactors such that <code class="code">g.gcd = g.coeff1 * <var class="Arg">m</var> + g.coeff2 * <var class="Arg">n</var></code>, and the components <code class="code">coeff3</code> and <code class="code">coeff4</code> are integer cofactors such that <code class="code">0 = g.coeff3 * <var class="Arg">m</var> + g.coeff4 * <var class="Arg">n</var></code>.</p>
<p>If <var class="Arg">m</var> and <var class="Arg">n</var> both are nonzero, <code class="code">AbsInt( g.coeff1 )</code> is less than or equal to <code class="code">AbsInt(<var class="Arg">n</var>) / (2 * g.gcd)</code>, and <code class="code">AbsInt( g.coeff2 )</code> is less than or equal to <code class="code">AbsInt(<var class="Arg">m</var>) / (2 * g.gcd)</code>.</p>
<p>If <var class="Arg">m</var> or <var class="Arg">n</var> or both are zero <code class="code">coeff3</code> is <code class="code">-<var class="Arg">n</var> / g.gcd</code> and <code class="code">coeff4</code> is <code class="code"><var class="Arg">m</var> / g.gcd</code>.</p>
<p>The coefficients always form a unimodular matrix, i.e., the determinant <code class="code">g.coeff1 * g.coeff4 - g.coeff3 * g.coeff2</code> is <span class="SimpleMath">1</span> or <span class="SimpleMath">-1</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Gcdex( 123, 66 );</span>
rec( coeff1 := 7, coeff2 := -13, coeff3 := -22, coeff4 := 41,
gcd := 3 )
</pre></div>
<p>This means <span class="SimpleMath">3 = 7 * 123 - 13 * 66</span>, <span class="SimpleMath">0 = -22 * 123 + 41 * 66</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Gcdex( 0, -3 );</span>
rec( coeff1 := 0, coeff2 := -1, coeff3 := 1, coeff4 := 0, gcd := 3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Gcdex( 0, 0 );</span>
rec( coeff1 := 1, coeff2 := 0, coeff3 := 0, coeff4 := 1, gcd := 0 )
</pre></div>
<p><code class="func">GcdRepresentation</code> (<a href="chap56.html#X7ABB91EF838075EF"><span class="RefLink">56.7-3</span></a>) provides similar functionality over arbitrary Euclidean rings.</p>
<p><a id="X7B33143E78A8DDE3" name="X7B33143E78A8DDE3"></a></p>
<h5>14.3-6 LcmInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LcmInt</code>( <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the least common multiple of the integers <var class="Arg">m</var> and <var class="Arg">n</var>.</p>
<p><code class="func">LcmInt</code> is a method used by the general operation <code class="func">Lcm</code> (<a href="chap56.html#X7ABA92057DD6C7AF"><span class="RefLink">56.7-6</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LcmInt( 123, 66 );</span>
2706
</pre></div>
<p><a id="X79B466E984CD52D4" name="X79B466E984CD52D4"></a></p>
<h5>14.3-7 CoefficientsQadic</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoefficientsQadic</code>( <var class="Arg">i</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the <var class="Arg">q</var>-adic representation of the integer <var class="Arg">i</var> as a list <span class="SimpleMath">l</span> of coefficients satisfying the equality <span class="SimpleMath"><var class="Arg">i</var> = ∑_{j = 0} <var class="Arg">q</var>^j ⋅ l[j+1]</span> for an integer <span class="SimpleMath"><var class="Arg">q</var> > 1</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=CoefficientsQadic(462,3);</span>
[ 0, 1, 0, 2, 2, 1 ]
</pre></div>
<p><a id="X83124F86839DC7E6" name="X83124F86839DC7E6"></a></p>
<h5>14.3-8 CoefficientsMultiadic</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoefficientsMultiadic</code>( <var class="Arg">ints</var>, <var class="Arg">int</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the multiadic expansion of the integer <var class="Arg">int</var> modulo the integers given in <var class="Arg">ints</var> (in ascending order). It returns a list of coefficients in the <em>reverse</em> order to that in <var class="Arg">ints</var>.</p>
<p><a id="X84A1900E82902B5F" name="X84A1900E82902B5F"></a></p>
<h5>14.3-9 ChineseRem</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChineseRem</code>( <var class="Arg">moduli</var>, <var class="Arg">residues</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ChineseRem</code> returns the combination of the <var class="Arg">residues</var> modulo the <var class="Arg">moduli</var>, i.e., the unique integer <code class="code">c</code> from <code class="code">[0..Lcm(<var class="Arg">moduli</var>)-1]</code> such that <code class="code">c = <var class="Arg">residues</var>[i]</code> modulo <code class="code"><var class="Arg">moduli</var>[i]</code> for all <code class="code">i</code>, if it exists. If no such combination exists <code class="func">ChineseRem</code> signals an error.</p>
<p>Such a combination does exist if and only if <code class="code"><var class="Arg">residues</var>[i] = <var class="Arg">residues</var>[k] mod Gcd( <var class="Arg">moduli</var>[i], <var class="Arg">moduli</var>[k] )</code> for every pair <code class="code">i</code>, <code class="code">k</code>. Note that this implies that such a combination exists if the moduli are pairwise relatively prime. This is called the Chinese remainder theorem.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ChineseRem( [ 2, 3, 5, 7 ], [ 1, 2, 3, 4 ] );</span>
53
<span class="GAPprompt">gap></span> <span class="GAPinput">ChineseRem( [ 6, 10, 14 ], [ 1, 3, 5 ] );</span>
103
</pre></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ChineseRem( [ 6, 10, 14 ], [ 1, 2, 3 ] );</span>
Error, the residues must be equal modulo 2 called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">gap> </span>
</pre></div>
<p><a id="X7E404B1183DBC82A" name="X7E404B1183DBC82A"></a></p>
<h5>14.3-10 PowerModInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PowerModInt</code>( <var class="Arg">r</var>, <var class="Arg">e</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <span class="SimpleMath"><var class="Arg">r</var>^<var class="Arg">e</var> mod <var class="Arg">m</var></span> for integers <var class="Arg">r</var>, <var class="Arg">e</var> and <var class="Arg">m</var>.</p>
<p>Note that <code class="func">PowerModInt</code> can reduce intermediate results and thus will generally be faster than using <var class="Arg">r</var><code class="code">^</code><var class="Arg">e</var><code class="code"> mod </code><var class="Arg">m</var>, which would compute <span class="SimpleMath"><var class="Arg">r</var>^<var class="Arg">e</var></span> first and reduces the result afterwards.</p>
<p><code class="func">PowerModInt</code> is a method for the general operation <code class="func">PowerMod</code> (<a href="chap56.html#X805A35D684B7A952"><span class="RefLink">56.7-9</span></a>).</p>
<p><a id="X82005E587F0CB02A" name="X82005E587F0CB02A"></a></p>
<h4>14.4 <span class="Heading">Prime Integers and Factorization</span></h4>
<p><a id="X86F5E4CD82FEB9F4" name="X86F5E4CD82FEB9F4"></a></p>
<h5>14.4-1 Primes</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Primes</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p><code class="func">Primes</code> is a strictly sorted list of the 168 primes less than 1000.</p>
<p>This is used in <code class="func">IsPrimeInt</code> (<a href="chap14.html#X78FDA4437EDCA70C"><span class="RefLink">14.4-2</span></a>) and <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>) to cast out small primes quickly.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Primes[1];</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Primes[100];</span>
541
</pre></div>
<p><a id="X78FDA4437EDCA70C" name="X78FDA4437EDCA70C"></a></p>
<h5>14.4-2 IsPrimeInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimeInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsProbablyPrimeInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IsPrimeInt</code> returns <code class="keyw">false</code> if it can prove that the integer <var class="Arg">n</var> is composite and <code class="keyw">true</code> otherwise. By convention <code class="code">IsPrimeInt(0) = IsPrimeInt(1) = false</code> and we define <code class="code">IsPrimeInt(-</code><var class="Arg">n</var><code class="code">) = IsPrimeInt(</code><var class="Arg">n</var><code class="code">)</code>.</p>
<p><code class="func">IsPrimeInt</code> will return <code class="keyw">true</code> for every prime <var class="Arg">n</var>. <code class="func">IsPrimeInt</code> will return <code class="keyw">false</code> for all composite <var class="Arg">n</var> <span class="SimpleMath">< 10^18</span> and for all composite <var class="Arg">n</var> that have a factor <span class="SimpleMath">p < 1000</span>. So for integers <var class="Arg">n</var> <span class="SimpleMath">< 10^18</span>, <code class="func">IsPrimeInt</code> is a proper primality test. It is conceivable that <code class="func">IsPrimeInt</code> may return <code class="keyw">true</code> for some composite <var class="Arg">n</var> <span class="SimpleMath">> 10^18</span>, but no such <var class="Arg">n</var> is currently known. So for integers <var class="Arg">n</var> <span class="SimpleMath">> 10^18</span>, <code class="func">IsPrimeInt</code> is a probable-primality test. <code class="func">IsPrimeInt</code> will issue a warning when its argument is probably prime but not a proven prime. (The function <code class="func">IsProbablyPrimeInt</code> will do a similar calculation but not issue a warning.) The warning can be switched off by <code class="code">SetInfoLevel( InfoPrimeInt, 0 );</code>, the default level is <span class="SimpleMath">1</span> (also see <code class="func">SetInfoLevel</code> (<a href="chap7.html#X7A43B9E68765EE9E"><span class="RefLink">7.4-3</span></a>) ).</p>
<p>If composites that fool <code class="func">IsPrimeInt</code> do exist, they would be extremely rare, and finding one by pure chance might be less likely than finding a bug in <strong class="pkg">GAP</strong>. We would appreciate being informed about any example of a composite number <var class="Arg">n</var> for which <code class="func">IsPrimeInt</code> returns <code class="keyw">true</code>.</p>
<p><code class="func">IsPrimeInt</code> is a deterministic algorithm, i.e., the computations involve no random numbers, and repeated calls will always return the same result. <code class="func">IsPrimeInt</code> first does trial divisions by the primes less than 1000. Then it tests that <var class="Arg">n</var> is a strong pseudoprime w.r.t. the base 2. Finally it tests whether <var class="Arg">n</var> is a Lucas pseudoprime w.r.t. the smallest quadratic nonresidue of <var class="Arg">n</var>. A better description can be found in the comment in the library file <code class="file">primality.gi</code>.</p>
<p>The time taken by <code class="func">IsPrimeInt</code> is approximately proportional to the third power of the number of digits of <var class="Arg">n</var>. Testing numbers with several hundreds digits is quite feasible.</p>
<p><code class="func">IsPrimeInt</code> is a method for the general operation <code class="func">IsPrime</code> (<a href="chap56.html#X7AA107AE7F79C6D8"><span class="RefLink">56.5-8</span></a>).</p>
<p>Remark: In future versions of <strong class="pkg">GAP</strong> we hope to change the definition of <code class="func">IsPrimeInt</code> to return <code class="keyw">true</code> only for proven primes (currently, we lack a sufficiently good primality proving function). In applications, use explicitly <code class="func">IsPrimeInt</code> or <code class="func">IsProbablyPrimeInt</code> with this change in mind.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimeInt( 2^31 - 1 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimeInt( 10^42 + 1 );</span>
false
</pre></div>
<p><a id="X7CD977B17B4A7A4B" name="X7CD977B17B4A7A4B"></a></p>
<h5>14.4-3 PrimalityProof</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimalityProof</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Construct a machine verifiable proof of the primality of (the probable prime) <var class="Arg">n</var>, following the ideas of <a href="chapBib.html#biBBLS1975">[BLS75]</a>. The proof consists of various Fermat and Lucas pseudoprimality tests, which taken as a whole prove the primality. The proof is represented as a list of witnesses of two kinds. The first kind, <code class="code">[ "F", divisor, base ]</code>, indicates a successful Fermat pseudoprimality test, where <var class="Arg">n</var> is a strong pseudoprime at <code class="keyw">base</code> with order not divisible by <span class="SimpleMath">(<var class="Arg">n</var>-1)/divisor</span>. The second kind, <code class="code">[ "L", divisor, discriminant, P ]</code> indicates a successful Lucas pseudoprimality test, for a quadratic form of given <code class="keyw">discriminant</code> and middle term <code class="keyw">P</code> with an extra check at <span class="SimpleMath">(<var class="Arg">n</var>+1)/divisor</span>.</p>
<p><a id="X8443125D7FD6F2A6" name="X8443125D7FD6F2A6"></a></p>
<h5>14.4-4 IsPrimePowerInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimePowerInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IsPrimePowerInt</code> returns <code class="keyw">true</code> if the integer <var class="Arg">n</var> is a prime power and <code class="keyw">false</code> otherwise.</p>
<p>An integer <span class="SimpleMath">n</span> is a <em>prime power</em> if there exists a prime <span class="SimpleMath">p</span> and a positive integer <span class="SimpleMath">i</span> such that <span class="SimpleMath">p^i = n</span>. If <span class="SimpleMath">n</span> is negative the condition is that there must exist a negative prime <span class="SimpleMath">p</span> and an odd positive integer <span class="SimpleMath">i</span> such that <span class="SimpleMath">p^i = n</span>. The integers 1 and -1 are not prime powers.</p>
<p>Note that <code class="func">IsPrimePowerInt</code> uses <code class="func">SmallestRootInt</code> (<a href="chap14.html#X7F98A0CE7B9FD366"><span class="RefLink">14.2-10</span></a>) and a probable-primality test (see <code class="func">IsPrimeInt</code> (<a href="chap14.html#X78FDA4437EDCA70C"><span class="RefLink">14.4-2</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimePowerInt( 31^5 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimePowerInt( 2^31-1 ); # 2^31-1 is actually a prime</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimePowerInt( 2^63-1 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered( [-10..10], IsPrimePowerInt );</span>
[ -8, -7, -5, -3, -2, 2, 3, 4, 5, 7, 8, 9 ]
</pre></div>
<p><a id="X78744C367A94C69F" name="X78744C367A94C69F"></a></p>
<h5>14.4-5 NextPrimeInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NextPrimeInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">NextPrimeInt</code> returns the smallest prime which is strictly larger than the integer <var class="Arg">n</var>.</p>
<p>Note that <code class="func">NextPrimeInt</code> uses a probable-primality test (see <code class="func">IsPrimeInt</code> (<a href="chap14.html#X78FDA4437EDCA70C"><span class="RefLink">14.4-2</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NextPrimeInt( 541 ); NextPrimeInt( -1 );</span>
547
2
</pre></div>
<p><a id="X819060E17E83728A" name="X819060E17E83728A"></a></p>
<h5>14.4-6 PrevPrimeInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrevPrimeInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PrevPrimeInt</code> returns the largest prime which is strictly smaller than the integer <var class="Arg">n</var>.</p>
<p>Note that <code class="func">PrevPrimeInt</code> uses a probable-primality test (see <code class="func">IsPrimeInt</code> (<a href="chap14.html#X78FDA4437EDCA70C"><span class="RefLink">14.4-2</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrevPrimeInt( 541 ); PrevPrimeInt( 1 );</span>
523
-2
</pre></div>
<p><a id="X82C989DB84744B36" name="X82C989DB84744B36"></a></p>
<h5>14.4-7 FactorsInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorsInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FactorsInt</code>( <var class="Arg">n:</var> <var class="Arg">RhoTrials</var> <var class="Arg">:=</var> <var class="Arg">trials</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">FactorsInt</code> returns a list of factors of a given integer <var class="Arg">n</var> such that <code class="code">Product( FactorsInt( <var class="Arg">n</var> ) ) = <var class="Arg">n</var></code>. If <span class="SimpleMath">|n| ≤ 1</span> the list <code class="code">[<var class="Arg">n</var>]</code> is returned. Otherwise the result contains probable primes, sorted by absolute value. The entries will all be positive except for the first one in case of a negative <var class="Arg">n</var>.</p>
<p>See <code class="func">PrimeDivisors</code> (<a href="chap14.html#X80E7A5D381C64CC9"><span class="RefLink">14.4-8</span></a>) for a function that returns a set of (probable) primes dividing <var class="Arg">n</var>.</p>
<p>Note that <code class="func">FactorsInt</code> uses a probable-primality test (see <code class="func">IsPrimeInt</code> (<a href="chap14.html#X78FDA4437EDCA70C"><span class="RefLink">14.4-2</span></a>)). Thus <code class="func">FactorsInt</code> might return a list which contains composite integers. In such a case you will get a warning about the use of a probable prime. You can switch off these warnings by <code class="code">SetInfoLevel( InfoPrimeInt, 0 );</code> (also see <code class="func">SetInfoLevel</code> (<a href="chap7.html#X7A43B9E68765EE9E"><span class="RefLink">7.4-3</span></a>)).</p>
<p>The time taken by <code class="func">FactorsInt</code> is approximately proportional to the square root of the second largest prime factor of <var class="Arg">n</var>, which is the last one that <code class="func">FactorsInt</code> has to find, since the largest factor is simply what remains when all others have been removed. Thus the time is roughly bounded by the fourth root of <var class="Arg">n</var>. <code class="func">FactorsInt</code> is guaranteed to find all factors less than <span class="SimpleMath">10^6</span> and will find most factors less than <span class="SimpleMath">10^10</span>. If <var class="Arg">n</var> contains multiple factors larger than that <code class="func">FactorsInt</code> may not be able to factor <var class="Arg">n</var> and will then signal an error.</p>
<p><code class="func">FactorsInt</code> is used in a method for the general operation <code class="func">Factors</code> (<a href="chap56.html#X82D6EDC685D12AE2"><span class="RefLink">56.5-9</span></a>).</p>
<p>In the second form, <code class="func">FactorsInt</code> calls <code class="code">FactorsRho</code> with a limit of <var class="Arg">trials</var> on the number of trials it performs. The default is 8192. Note that Pollard's Rho is the fastest method for finding prime factors with roughly 5-10 decimal digits, but becomes more and more inferior to other factorization techniques like e.g. the Elliptic Curves Method (ECM) the bigger the prime factors are. Therefore instead of performing a huge number of Rho <var class="Arg">trials</var>, it is usually more advisable to install the <strong class="pkg">FactInt</strong> package and then simply to use the operation <code class="func">Factors</code> (<a href="chap56.html#X82D6EDC685D12AE2"><span class="RefLink">56.5-9</span></a>). The factorization of the 8-th Fermat number by Pollard's Rho below takes already a while.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorsInt( -Factorial(6) );</span>
[ -2, 2, 2, 2, 3, 3, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( FactorsInt( Factorial(13)/11 ) );</span>
[ 2, 3, 5, 7, 13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorsInt( 2^63 - 1 );</span>
[ 7, 7, 73, 127, 337, 92737, 649657 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorsInt( 10^42 + 1 );</span>
[ 29, 101, 281, 9901, 226549, 121499449, 4458192223320340849 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FactorsInt(2^256+1:RhoTrials:=100000000);</span>
[ 1238926361552897,
93461639715357977769163558199606896584051237541638188580280321 ]
</pre></div>
<p><a id="X80E7A5D381C64CC9" name="X80E7A5D381C64CC9"></a></p>
<h5>14.4-8 PrimeDivisors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimeDivisors</code>( <var class="Arg">n</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">PrimeDivisors</code> returns for a non-zero integer <var class="Arg">n</var> a set of its positive (probable) primes divisors. In rare cases the result could contain a composite number which passed certain primality tests, see <code class="func">IsProbablyPrimeInt</code> (<a href="chap14.html#X78FDA4437EDCA70C"><span class="RefLink">14.4-2</span></a>) and <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>) for more details.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimeDivisors(-12);</span>
[ 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimeDivisors(1);</span>
[ ]
</pre></div>
<p><a id="X786FF92C7C54BF97" name="X786FF92C7C54BF97"></a></p>
<h5>14.4-9 PartialFactorization</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialFactorization</code>( <var class="Arg">n</var>[, <var class="Arg">effort</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">PartialFactorization</code> returns a partial factorization of the integer <var class="Arg">n</var>. No assertions are made about the primality of the factors, except of those mentioned below.</p>
<p>The argument <var class="Arg">effort</var>, if given, specifies how intensively the function should try to determine factors of <var class="Arg">n</var>. The default is <var class="Arg">effort</var> = 5.</p>
<ul>
<li><p>If <var class="Arg">effort</var> = 0, trial division by the primes below 100 is done. Returned factors below <span class="SimpleMath">10^4</span> are guaranteed to be prime.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 1, trial division by the primes below 1000 is done. Returned factors below <span class="SimpleMath">10^6</span> are guaranteed to be prime.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 2, additionally trial division by the numbers in the lists <code class="code">Primes2</code> and <code class="code">ProbablePrimes2</code> is done, and perfect powers are detected. Returned factors below <span class="SimpleMath">10^6</span> are guaranteed to be prime.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 3, additionally <code class="code">FactorsRho</code> (Pollard's Rho) with <code class="code">RhoTrials</code> = 256 is used.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 4, as above, but <code class="code">RhoTrials</code> = 2048.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 5, as above, but <code class="code">RhoTrials</code> = 8192. Returned factors below <span class="SimpleMath">10^12</span> are guaranteed to be prime, and all prime factors below <span class="SimpleMath">10^6</span> are guaranteed to be found.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 6 and the package <strong class="pkg">FactInt</strong> is loaded, in addition to the above quite a number of special cases are handled.</p>
</li>
<li><p>If <var class="Arg">effort</var> = 7 and the package <strong class="pkg">FactInt</strong> is loaded, the only thing which is not attempted to obtain a full factorization into Baillie-Pomerance-Selfridge-Wagstaff pseudoprimes is the application of the MPQS to a remaining composite with more than 50 decimal digits.</p>
</li>
</ul>
<p>Increasing the value of the argument <var class="Arg">effort</var> by one usually results in an increase of the runtime requirements by a factor of (very roughly!) 3 to 10. (Also see <code class="func">CheapFactorsInt</code> (<span class="RefLink">???</span>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..5],i->PartialFactorization(97^35-1,i)); </span>
[ [ 2, 2, 2, 2, 2, 3, 11, 31, 43,
2446338959059521520901826365168917110105972824229555319002965029 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967,
2529823122088440042297648774735177983563570655873376751812787 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967,
2529823122088440042297648774735177983563570655873376751812787 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321,
242549173950325921859769421435653153445616962914227 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, 687121,
352993394104278463123335513593170858474150787 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, 687121,
20241187, 504769301, 34549173843451574629911361501 ] ]
</pre></div>
<p><a id="X803D431087B6FF28" name="X803D431087B6FF28"></a></p>
<h5>14.4-10 PrintFactorsInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrintFactorsInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints the prime factorization of the integer <var class="Arg">n</var> in human-readable form. See also <code class="func">StringPP</code> (<a href="chap27.html#X7BB1059185AB4F84"><span class="RefLink">27.7-9</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrintFactorsInt( Factorial( 7 ) ); Print( "\n" );</span>
2^4*3^2*5*7
</pre></div>
<p><a id="X82148B347E294C87" name="X82148B347E294C87"></a></p>
<h5>14.4-11 PrimePowersInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimePowersInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the prime factorization of the integer <var class="Arg">n</var> as a list <span class="SimpleMath">[ p_1, e_1, ..., p_k, e_k ]</span> with <var class="Arg">n</var> = <span class="SimpleMath">p_1^{e_1} ⋅ p_2^{e_2} ⋅ ... ⋅ p_k^{e_k}</span>.</p>
<p>For negative integers, the absolute value is taken. Zero is not allowed as input.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimePowersInt( Factorial( 7 ) );</span>
[ 2, 4, 3, 2, 5, 1, 7, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimePowersInt( 1 );</span>
[ ]
</pre></div>
<p><a id="X809E0E1B83AF7695" name="X809E0E1B83AF7695"></a></p>
<h5>14.4-12 DivisorsInt</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DivisorsInt</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">DivisorsInt</code> returns a list of all divisors of the integer <var class="Arg">n</var>. The list is sorted, so that it starts with 1 and ends with <var class="Arg">n</var>. We define that <code class="code">DivisorsInt( -<var class="Arg">n</var> ) = DivisorsInt( <var class="Arg">n</var> )</code>.</p>
<p>Since the set of divisors of 0 is infinite calling <code class="code">DivisorsInt( 0 )</code> causes an error.</p>
<p><code class="func">DivisorsInt</code> may call <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>) to obtain the prime factors. <code class="func">Sigma</code> (<a href="chap15.html#X823707DF821E79A0"><span class="RefLink">15.5-1</span></a>) and <code class="func">Tau</code> (<a href="chap15.html#X798C62847EE0372E"><span class="RefLink">15.5-2</span></a>) compute the sum and the number of positive divisors, respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DivisorsInt( 1 ); DivisorsInt( 20 ); DivisorsInt( 541 );</span>
[ 1 ]
[ 1, 2, 4, 5, 10, 20 ]
[ 1, 541 ]
</pre></div>
<p><a id="X864BF040862409FC" name="X864BF040862409FC"></a></p>
<h4>14.5 <span class="Heading">Residue Class Rings</span></h4>
<p><code class="func">ZmodnZ</code> (<a href="chap14.html#X79CE76AD82B3E2B2"><span class="RefLink">14.5-2</span></a>) returns a residue class ring of <code class="func">Integers</code> (<a href="chap14.html#X853DF11B80068ED5"><span class="RefLink">14</span></a>) modulo an ideal. These residue class rings are rings, thus all operations for rings (see Chapter <a href="chap56.html#X81897F6082CACB59"><span class="RefLink">56</span></a>) apply. See also Chapters <a href="chap59.html#X7893ABF67A028802"><span class="RefLink">59</span></a> and <a href="chap15.html#X7FB995737B7ED8A2"><span class="RefLink">15</span></a>.</p>
<p><a id="X87B1210B8581D5B2" name="X87B1210B8581D5B2"></a></p>
<h5><code>14.5-1 \mod</code></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \mod</code>( <var class="Arg">r/s</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">r</var>, <var class="Arg">s</var> and <var class="Arg">n</var> are integers, <code class="code"><var class="Arg">r</var> / <var class="Arg">s</var></code> as a reduced fraction is <code class="code">p/q</code>, where <code class="code">q</code> and <var class="Arg">n</var> are coprime, then <code class="code"><var class="Arg">r</var> / <var class="Arg">s</var> mod <var class="Arg">n</var></code> is defined to be the product of <code class="code">p</code> and the inverse of <code class="code">q</code> modulo <var class="Arg">n</var>. See Section <a href="chap4.html#X7B66C8707B5DE10A"><span class="RefLink">4.13</span></a> for more details and definitions.</p>
<p>With the above definition, <code class="code">4 / 6 mod 32</code> equals <code class="code">2 / 3 mod 32</code> and hence exists (and is equal to 22), despite the fact that 6 has no inverse modulo 32.</p>
<p><a id="X79CE76AD82B3E2B2" name="X79CE76AD82B3E2B2"></a></p>
<h5>14.5-2 ZmodnZ</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZmodnZ</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZmodpZ</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZmodpZNC</code>( <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">ZmodnZ</code> returns a ring <span class="SimpleMath">R</span> isomorphic to the residue class ring of the integers modulo the ideal generated by <var class="Arg">n</var>. The element corresponding to the residue class of the integer <span class="SimpleMath">i</span> in this ring can be obtained by <code class="code">i * One( R )</code>, and a representative of the residue class corresponding to the element <span class="SimpleMath">x ∈ R</span> can be computed by <code class="code">Int</code><span class="SimpleMath">( x )</span>.</p>
<p><code class="code">ZmodnZ( <var class="Arg">n</var> )</code> is equal to <code class="code">Integers mod <var class="Arg">n</var></code>.</p>
<p><code class="func">ZmodpZ</code> does the same if the argument <var class="Arg">p</var> is a prime integer, additionally the result is a field. <code class="func">ZmodpZNC</code> omits the check whether <var class="Arg">p</var> is a prime.</p>
<p>Each ring returned by these functions contains the whole family of its elements if <var class="Arg">n</var> is not a prime, and is embedded into the family of finite field elements of characteristic <var class="Arg">n</var> if <var class="Arg">n</var> is a prime.</p>
<p><a id="X838F36507D985EDA" name="X838F36507D985EDA"></a></p>
<h5>14.5-3 ZmodnZObj</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZmodnZObj</code>( <var class="Arg">Fam</var>, <var class="Arg">r</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZmodnZObj</code>( <var class="Arg">r</var>, <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If the first argument is a residue class family <var class="Arg">Fam</var> then <code class="func">ZmodnZObj</code> returns the element in <var class="Arg">Fam</var> whose coset is represented by the integer <var class="Arg">r</var>.</p>
<p>If the two arguments are an integer <var class="Arg">r</var> and a positive integer <var class="Arg">n</var> then <code class="func">ZmodnZObj</code> returns the element in <code class="code">ZmodnZ( <var class="Arg">n</var> )</code> (see <code class="func">ZmodnZ</code> (<a href="chap14.html#X79CE76AD82B3E2B2"><span class="RefLink">14.5-2</span></a>)) whose coset is represented by the integer <var class="Arg">r</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= ZmodnZ(15);</span>
(Integers mod 15)
<span class="GAPprompt">gap></span> <span class="GAPinput">fam:=ElementsFamily(FamilyObj(r));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:= ZmodnZObj(fam,9);</span>
ZmodnZObj( 9, 15 )
<span class="GAPprompt">gap></span> <span class="GAPinput">a+a;</span>
ZmodnZObj( 3, 15 )
<span class="GAPprompt">gap></span> <span class="GAPinput">Int(a+a);</span>
3
</pre></div>
<p><a id="X7D0107DD79753901" name="X7D0107DD79753901"></a></p>
<h5>14.5-4 IsZmodnZObj</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZmodnZObj</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZmodnZObjNonprime</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZmodpZObj</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZmodpZObjSmall</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsZmodpZObjLarge</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>The elements in the rings <span class="SimpleMath">Z / n Z</span> are in the category <code class="func">IsZmodnZObj</code>. If <span class="SimpleMath">n</span> is a prime then the elements are of course also in the category <code class="func">IsFFE</code> (<a href="chap59.html#X7D3DF32C84FEBD25"><span class="RefLink">59.1-1</span></a>), otherwise they are in <code class="func">IsZmodnZObjNonprime</code>. <code class="func">IsZmodpZObj</code> is an abbreviation of <code class="code">IsZmodnZObj and IsFFE</code>. This category is the disjoint union of <code class="func">IsZmodpZObjSmall</code> and <code class="func">IsZmodpZObjLarge</code>, the former containing all elements with <span class="SimpleMath">n</span> at most <code class="code">MAXSIZE_GF_INTERNAL</code>.</p>
<p>The reasons to distinguish the prime case from the nonprime case are</p>
<ul>
<li><p>that objects in <code class="func">IsZmodnZObjNonprime</code> have an external representation (namely the residue in the range <span class="SimpleMath">[ 0, 1, ..., n-1 ]</span>),</p>
</li>
<li><p>that the comparison of elements can be defined as comparison of the residues, and</p>
</li>
<li><p>that the elements lie in a family of type <code class="code">IsZmodnZObjNonprimeFamily</code> (note that for prime <span class="SimpleMath">n</span>, the family must be an <code class="code">IsFFEFamily</code>).</p>
</li>
</ul>
<p>The reasons to distinguish the small and the large case are that for small <span class="SimpleMath">n</span> the elements must be compatible with the internal representation of finite field elements, whereas we are free to define comparison as comparison of residues for large <span class="SimpleMath">n</span>.</p>
<p>Note that we <em>cannot</em> claim that every finite field element of degree 1 is in <code class="func">IsZmodnZObj</code>, since finite field elements in internal representation may not know that they lie in the prime field.</p>
<p><a id="X7904B6D681EBF091" name="X7904B6D681EBF091"></a></p>
<h4>14.6 <span class="Heading">Check Digits</span></h4>
<p><a id="X82BABA8F868BD425" name="X82BABA8F868BD425"></a></p>
<h5>14.6-1 CheckDigitISBN</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CheckDigitISBN</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CheckDigitISBN13</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CheckDigitPostalMoneyOrder</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CheckDigitUPC</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>These functions can be used to compute, or check, check digits for some everyday items. In each case what is submitted as input is either the number with check digit (in which case the function returns <code class="code">true</code> or <code class="code">false</code>), or the number without check digit (in which case the function returns the missing check digit). The number can be specified as integer, as string (for example in case of leading zeros) or as a sequence of arguments, each representing a single digit. The check digits tested are the 10-digit ISBN (International Standard Book Number) using <code class="func">CheckDigitISBN</code> (since arithmetic is module 11, a digit 11 is represented by an X); the newer 13-digit ISBN-13 using <code class="func">CheckDigitISBN13</code>; the numbers of 11-digit US postal money orders using <code class="func">CheckDigitPostalMoneyOrder</code>; and the 12-digit UPC bar code found on groceries using <code class="func">CheckDigitUPC</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitISBN("052166103");</span>
Check Digit is 'X'
'X'
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitISBN("052166103X");</span>
Checksum test satisfied
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitISBN(0,5,2,1,6,6,1,0,3,1);</span>
Checksum test failed
false
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitISBN(0,5,2,1,6,6,1,0,3,'X'); # note single quotes!</span>
Checksum test satisfied
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitISBN13("9781420094527");</span>
Checksum test satisfied
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitUPC("07164183001");</span>
Check Digit is 1
1
<span class="GAPprompt">gap></span> <span class="GAPinput">CheckDigitPostalMoneyOrder(16786457155);</span>
Checksum test satisfied
true
</pre></div>
<p><a id="X85F1A6A5870485B9" name="X85F1A6A5870485B9"></a></p>
<h5>14.6-2 CheckDigitTestFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CheckDigitTestFunction</code>( <var class="Arg">l</var>, <var class="Arg">m</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function creates check digit test functions such as <code class="func">CheckDigitISBN</code> (<a href="chap14.html#X82BABA8F868BD425"><span class="RefLink">14.6-1</span></a>) for check digit schemes that use the inner products with a fixed vector modulo a number. The scheme creates will use strings of <var class="Arg">l</var> digits (including the check digits), the check consists of taking the standard product of the vector of digits with the fixed vector <var class="Arg">f</var> modulo <var class="Arg">m</var>; the result needs to be 0. The function returns a function that then can be used for testing or determining check digits.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">isbntest:=CheckDigitTestFunction(10,11,[1,2,3,4,5,6,7,8,9,-1]); </span>
function( arg... ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">isbntest("038794680");</span>
Check Digit is 2
2
</pre></div>
<p><a id="X85361FAE8088C006" name="X85361FAE8088C006"></a></p>
<h4>14.7 <span class="Heading">Random Sources</span></h4>
<p><strong class="pkg">GAP</strong> provides <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>) methods for many collections of objects. On a lower level these methods use <em>random sources</em> which provide random integers and random choices from lists.</p>
<p><a id="X82E31A697E389F1D" name="X82E31A697E389F1D"></a></p>
<h5>14.7-1 IsRandomSource</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRandomSource</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>This is the category of random source objects which are defined to have, for an object <var class="Arg">rs</var> in this category, methods available for the following operations which are explained in more detail below: <code class="code">Random( <var class="Arg">rs</var>, <var class="Arg">list</var> )</code> giving a random element of a list, <code class="code">Random( <var class="Arg">rs</var>, <var class="Arg">low</var>, <var class="Arg">high</var> )</code> giving a random integer between <var class="Arg">low</var> and <var class="Arg">high</var> (inclusive), <code class="func">Init</code> (<a href="chap14.html#X819E3E3080297347"><span class="RefLink">14.7-3</span></a>), <code class="func">State</code> (<a href="chap14.html#X819E3E3080297347"><span class="RefLink">14.7-3</span></a>) and <code class="func">Reset</code> (<a href="chap14.html#X819E3E3080297347"><span class="RefLink">14.7-3</span></a>).</p>
<p>Use <code class="func">RandomSource</code> (<a href="chap14.html#X7CB0B5BC82F8FD8F"><span class="RefLink">14.7-5</span></a>) to construct new random sources.</p>
<p>One idea behind providing several independent (pseudo) random sources is to make algorithms which use some sort of random choices deterministic. They can use their own new random source created with a fixed seed and so do exactly the same in different calls.</p>
<p>Random source objects lie in the family <code class="code">RandomSourcesFamily</code>.</p>
<p><a id="X821004F286282D49" name="X821004F286282D49"></a></p>
<h5>14.7-2 Random</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">rs</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">rs</var>, <var class="Arg">low</var>, <var class="Arg">high</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This operation returns a random element from list <var class="Arg">list</var>, or an integer in the range from the given (possibly large) integers <var class="Arg">low</var> to <var class="Arg">high</var>, respectively.</p>
<p>The choice should only depend on the random source <var class="Arg">rs</var> and have no effect on other random sources.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mysource := RandomSource(IsMersenneTwister, 42);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(mysource, 1, 10^60);</span>
999331861769949319194941485000557997842686717712198687315183
</pre></div>
<p><a id="X819E3E3080297347" name="X819E3E3080297347"></a></p>
<h5>14.7-3 State</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ State</code>( <var class="Arg">rs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Reset</code>( <var class="Arg">rs</var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Init</code>( <var class="Arg">prers</var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>These are the basic operations for which random sources (see <code class="func">IsRandomSource</code> (<a href="chap14.html#X82E31A697E389F1D"><span class="RefLink">14.7-1</span></a>)) must have methods.</p>
<p><code class="func">State</code> should return a data structure which allows to recover the state of the random source such that a sequence of random calls using this random source can be reproduced. If a random source cannot be reset (say, it uses truly random physical data) then <code class="func">State</code> should return <code class="keyw">fail</code>.</p>
<p><code class="code">Reset( <var class="Arg">rs</var>, <var class="Arg">seed</var> )</code> resets the random source <var class="Arg">rs</var> to a state described by <var class="Arg">seed</var>, if the random source can be reset (otherwise it should do nothing). Here <var class="Arg">seed</var> can be an output of <code class="func">State</code> and then should reset to that state. Also, the methods should always allow integers as <var class="Arg">seed</var>. Without the <var class="Arg">seed</var> argument the default <span class="SimpleMath"><var class="Arg">seed</var> = 1</span> is used.</p>
<p><code class="func">Init</code> is the constructor of a random source, it gets an empty component object <var class="Arg">prers</var> which has already the correct type and should fill in the actual data which are needed. Optionally, it should allow one to specify a <var class="Arg">seed</var> for the initial state, as explained for <code class="func">Reset</code>.</p>
<p>Most methods for <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>) in the <strong class="pkg">GAP</strong> library use the <code class="func">GlobalMersenneTwister</code> (<a href="chap14.html#X7F772E2686B35865"><span class="RefLink">14.7-4</span></a>) as random source. It can be reset into a known state as in the following example.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">seed := Reset(GlobalMersenneTwister);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">seed = State(GlobalMersenneTwister);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..10],i->Random(Integers));</span>
[ -3, 2, -1, -2, -1, -1, 1, -4, 1, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..10],i->Random(Integers));</span>
[ -1, -1, -1, 1, -1, 1, -2, -1, -2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Reset(GlobalMersenneTwister, seed);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..10],i->Random(Integers));</span>
[ -3, 2, -1, -2, -1, -1, 1, -4, 1, 0 ]
</pre></div>
<p><a id="X7F772E2686B35865" name="X7F772E2686B35865"></a></p>
<h5>14.7-4 IsMersenneTwister</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMersenneTwister</code>( <var class="Arg">rs</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGAPRandomSource</code>( <var class="Arg">rs</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGlobalRandomSource</code>( <var class="Arg">rs</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GlobalMersenneTwister</code></td><td class="tdright">( global variable )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GlobalRandomSource</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Currently, the <strong class="pkg">GAP</strong> library provides three types of random sources, distinguished by the three listed categories.</p>
<p><code class="func">IsMersenneTwister</code> are random sources which use a fast random generator of 32 bit numbers, called the Mersenne twister. The pseudo random sequence has a period of <span class="SimpleMath">2^19937-1</span> and the numbers have a <span class="SimpleMath">623</span>-dimensional equidistribution. For more details and the origin of the code used in the <strong class="pkg">GAP</strong> kernel, see: <span class="URL"><a href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html">http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html</a></span>.</p>
<p>Use the Mersenne twister if possible, in particular for generating many large random integers.</p>
<p>There is also a predefined global random source <code class="func">GlobalMersenneTwister</code> which is used by most of the library methods for <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>).</p>
<p><code class="func">IsGAPRandomSource</code> uses the same number generator as <code class="func">IsGlobalRandomSource</code>, but you can create several of these random sources which generate their random numbers independently of all other random sources.</p>
<p><code class="func">IsGlobalRandomSource</code> gives access to the <em>classical</em> global random generator which was used by <strong class="pkg">GAP</strong> in former releases. You do not need to construct new random sources of this kind which would all use the same global data structure. Just use the existing random source <code class="func">GlobalRandomSource</code>. This uses the additive random number generator described in <a href="chapBib.html#biBTACP2">[Knu98]</a> (Algorithm A in 3.2.2 with lag <span class="SimpleMath">30</span>).</p>
<p><a id="X7CB0B5BC82F8FD8F" name="X7CB0B5BC82F8FD8F"></a></p>
<h5>14.7-5 RandomSource</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomSource</code>( <var class="Arg">cat</var>[, <var class="Arg">seed</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This operation is used to create new random sources. The first argument <var class="Arg">cat</var> is the category describing the type of the random generator, an optional <var class="Arg">seed</var> which can be an integer or a type specific data structure can be given to specify the initial state.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">rs1 := RandomSource(IsMersenneTwister);</span>
<RandomSource in IsMersenneTwister>
<span class="GAPprompt">gap></span> <span class="GAPinput">state1 := State(rs1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l1 := List([1..10000], i-> Random(rs1, [1..6]));; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rs2 := RandomSource(IsMersenneTwister);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l2 := List([1..10000], i-> Random(rs2, [1..6]));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l1 = l2;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l1 = List([1..10000], i-> Random(rs1, [1..6])); </span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">n := Random(rs1, 1, 2^220);</span>
1077726777923092117987668044202944212469136000816111066409337432400
</pre></div>
<p><a id="X7A0311DF78DB4FD8" name="X7A0311DF78DB4FD8"></a></p>
<h4>14.8 <span class="Heading">Bitfields</span></h4>
<p>Bitfields are a low-level feature intended to support efficient subdivision of immediate integers into bitfields of various widths. This is typically useful in implementing space-efficient and/or cache-efficient data structures. This feature should be used with care because (<em>inter alia</em>) it has different limitations on 32-bit and 64-bit architectures.</p>
<p><a id="X85C7BD9E7FCC6C10" name="X85C7BD9E7FCC6C10"></a></p>
<h5>14.8-1 MakeBitfields</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MakeBitfields</code>( <var class="Arg">width....</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function sets up the machinery for a set of bitfields of the given widths. All bitfield values are treated as unsigned. The total of the widths must not exceed 60 bits on 64-bit architecture or 28 bits on a 32-bit architecture. For performance reasons some checks that one might wish to do are ommitted. In particular, the builder and setter functions do not check if the value[s] passed to them are negative or too large (unless <strong class="pkg">GAP</strong> is specially compiled for debugging). Behaviour when such arguments are passed is undefined. You can tell which type of architecture you are running on by acccessing <code class="code">GAPInfo.BytesPerVariable</code> which is 8 on 64-bits and 4 on 32. The return value when <span class="SimpleMath">n</span> widths are given is a record whose fields are</p>
<dl>
<dt><strong class="Mark"><code class="code">widths</code></strong></dt>
<dd><p>a copy of the arguments, for convenience,</p>
</dd>
<dt><strong class="Mark"><code class="code">getters</code></strong></dt>
<dd><p>a list of <span class="SimpleMath">n</span> functions of one argument each of which extracts one of the fields from an immediate integer</p>
</dd>
<dt><strong class="Mark"><code class="code">setters</code></strong></dt>
<dd><p>a list of <span class="SimpleMath">n</span> functions each taking two arguments: a packed value and a new value for one of its fields and returning a new packed value. The <span class="SimpleMath">i</span>th function returned the new packed value in which the <span class="SimpleMath">i</span>th field has been replaced by the new value. Note that this does NOT modify the original packed value.</p>
</dd>
</dl>
<p>Two additional fields may be present if any of the field widths is one. Each is a list and only has entried bound in the positions corresponding to the width 1 fields.</p>
<dl>
<dt><strong class="Mark"><code class="code">booleanGetters</code></strong></dt>
<dd><p>if the <span class="SimpleMath">i</span>th position of this list is set, it contains a function which extracts the <span class="SimpleMath">i</span>th field (which will have width one) and returns <code class="code">true</code> if it contains 1 and <code class="code">false</code> if it contains 0</p>
</dd>
<dt><strong class="Mark"><code class="code">booleanSetters</code></strong></dt>
<dd><p>if the <span class="SimpleMath">i</span>th position of this list is set, it contains a function of two arguments. The first argument is a packed value, the second is <code class="code">true</code> or <code class="code">false</code>. It returns a new packed value in which the <span class="SimpleMath">i</span>th field is set to 1 if the second argument was <code class="code">true</code> and 0 if it was <code class="code">false</code>. Behaviour for any other value is undefined.</p>
</dd>
</dl>
<p><a id="X8068CE3781F4003C" name="X8068CE3781F4003C"></a></p>
<h5>14.8-2 BuildBitfields</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BuildBitfields</code>( <var class="Arg">widths</var>, <var class="Arg">vals...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function takes one or more argument. It's first argument is a list of field widths, as found in the <code class="code">widths</code> entry of a record returned by <code class="code">MakeBitfields</code>. The remaining arguments are unsigned integer values, equal in number to the entries of the list of field widths. It returns a small integer in which those entries are packed into bitfields of the given widths. The first entry occupies the least significant bits. DeclareGlobalFunction("BuildBitfields");</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">bf := MakeBitfields(1,2,3);</span>
rec( booleanGetters := [ function( data ) ... end ],
booleanSetters := [ function( data, val ) ... end ],
getters := [ function( data ) ... end, function( data ) ... end,
function( data ) ... end ],
setters := [ function( data, val ) ... end, function( data, val ) ... end,
function( data, val ) ... end ], widths := [ 1, 2, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">x := BuildBitfields(bf.widths,0,3,5);</span>
46
<span class="GAPprompt">gap></span> <span class="GAPinput">bf.getters[3](x);</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">y := bf.setters[1](x,1);</span>
47
<span class="GAPprompt">gap></span> <span class="GAPinput">x;</span>
46
<span class="GAPprompt">gap></span> <span class="GAPinput">bf.booleanGetters[1](x);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">bf.booleanGetters[1](y);</span>
true
</pre></div>
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