<html><head><title>[ref] 14 Integers</title></head>
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<h1>14 Integers</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP014.htm#SECT001">Elementary Operations for Integers</a>
<li> <A HREF="CHAP014.htm#SECT002">Quotients and Remainders</a>
<li> <A HREF="CHAP014.htm#SECT003">Prime Integers and Factorization</a>
<li> <A HREF="CHAP014.htm#SECT004">Residue Class Rings</a>
<li> <A HREF="CHAP014.htm#SECT005">Random Sources</a>
</ol><p>
<p>
One of the most fundamental datatypes in every programming language is
the integer type. <font face="Gill Sans,Helvetica,Arial">GAP</font> is no exception.
<p>
<font face="Gill Sans,Helvetica,Arial">GAP</font> integers are entered as a sequence of decimal digits
optionally preceded by a <code>+</code> sign for positive integers or a <code>-</code> sign for
negative integers.
The size of integers in <font face="Gill Sans,Helvetica,Arial">GAP</font> is only limited by the amount of available
memory, so you can compute with integers having thousands of digits.
<p>
<pre>
gap> -1234;
-1234
gap> 123456789012345678901234567890123456789012345678901234567890;
123456789012345678901234567890123456789012345678901234567890
</pre>
<p>
Many more functions that are mainly related to the prime residue group of
integers modulo an integer are described in chapter <a href="CHAP015.htm">Number Theory</a>,
and functions dealing with combinatorics can be found
in chapter <a href="CHAP017.htm">Combinatorics</a>.
<p>
<a name = ""></a>
<li><code>Integers V</code>
<a name = ""></a>
<li><code>PositiveIntegers V</code>
<a name = ""></a>
<li><code>NonnegativeIntegers V</code>
<p>
These global variables represent the ring of integers and the semirings
of positive and nonnegative integers, respectively.
<p>
<pre>
gap> Size( Integers ); 2 in Integers;
infinity
true
</pre>
<p>
<a name = ""></a>
<li><code>IsIntegers( </code><var>obj</var><code> ) C</code>
<a name = ""></a>
<li><code>IsPositiveIntegers( </code><var>obj</var><code> ) C</code>
<a name = ""></a>
<li><code>IsNonnegativeIntegers( </code><var>obj</var><code> ) C</code>
<p>
are the defining categories for the domains <code>Integers</code>,
<code>PositiveIntegers</code>, and <code>NonnegativeIntegers</code>.
<p>
<pre>
gap> IsIntegers( Integers ); IsIntegers( Rationals ); IsIntegers( 7 );
true
false
false
</pre>
<p>
<code>Integers</code> is a subset of <code>Rationals</code>, which is a subset of <code>Cyclotomics</code>.
See Chapter <a href="CHAP018.htm">Cyclotomic Numbers</a> for arithmetic operations and comparison of
integers.
<p>
<p>
<h2><a name="SECT001">14.1 Elementary Operations for Integers</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>IsInt( </code><var>obj</var><code> ) C</code>
<p>
Every rational integer lies in the category <code>IsInt</code>,
which is a subcategory of <code>IsRat</code> (see <a href="CHAP016.htm">Rational Numbers</a>).
<p>
<a name = "SSEC001.2"></a>
<li><code>IsPosInt( </code><var>obj</var><code> ) C</code>
<p>
Every positive integer lies in the category <code>IsPosInt</code>.
<p>
<a name = "SSEC001.3"></a>
<li><code>Int( </code><var>elm</var><code> ) A</code>
<p>
<code>Int</code> returns an integer <var>int</var> whose meaning depends on the type
of <var>elm</var>.
<p>
If <var>elm</var> is a rational number (see <a href="CHAP016.htm">Rational Numbers</a>) then <var>int</var> is the
integer part of the quotient of numerator and denominator of <var>elm</var>
(see <a href="CHAP014.htm#SSEC002.1">QuoInt</a>).
<p>
If <var>elm</var> is an element of a finite prime field
(see Chapter <a href="CHAP057.htm">Finite Fields</a>) then <var>int</var> is the smallest
nonnegative integer such that <code></code><var>elm</var><code> = </code><var>int</var><code> * One( </code><var>elm</var><code> )</code>.
<p>
If <var>elm</var> is a string (see Chapter <a href="CHAP026.htm">Strings and Characters</a>) consisting of
digits <code>'0'</code>, <code>'1'</code>, …, <code>'9'</code>
and <code>'-'</code> (at the first position) then <var>int</var> is the integer
described by this string.
The operation <code>String</code> (see <a href="CHAP026.htm#SSEC005.1">String</a>) can be used to compute a string for
rational integers, in fact for all cyclotomics.
<p>
<pre>
gap> Int( 4/3 ); Int( -2/3 );
1
0
gap> int:= Int( Z(5) ); int * One( Z(5) );
2
Z(5)
gap> Int( "12345" ); Int( "-27" ); Int( "-27/3" );
12345
-27
fail
</pre>
<p>
<a name = "SSEC001.4"></a>
<li><code>IsEvenInt( </code><var>n</var><code> ) F</code>
<p>
tests if the integer <var>n</var> is divisible by 2.
<p>
<a name = "SSEC001.5"></a>
<li><code>IsOddInt( </code><var>n</var><code> ) F</code>
<p>
tests if the integer <var>n</var> is not divisible by 2.
<p>
<a name = "SSEC001.6"></a>
<li><code>AbsInt( </code><var>n</var><code> ) F</code>
<p>
<code>AbsInt</code> returns the absolute value of the integer <var>n</var>, i.e., <var>n</var> if <var>n</var>
is positive, -<var>n</var> if <var>n</var> is negative and 0 if <var>n</var> is 0.
<p>
<code>AbsInt</code> is a special case of the general operation <code>EuclideanDegree</code>
see <a href="CHAP054.htm#SSEC006.2">EuclideanDegree</a>).
<p>
<a name = "I0"></a>
See also <a href="CHAP018.htm#SSEC001.6">AbsoluteValue</a>.
<pre>
gap> AbsInt( 33 );
33
gap> AbsInt( -214378 );
214378
gap> AbsInt( 0 );
0
</pre>
<p>
<a name = "SSEC001.7"></a>
<li><code>SignInt( </code><var>n</var><code> ) F</code>
<p>
<code>SignInt</code> returns the sign of the integer <var>n</var>, i.e., 1 if <var>n</var> is
positive, -1 if <var>n</var> is negative and 0 if <var>n</var> is 0.
<p>
<a name = "I1"></a>
<pre>
gap> SignInt( 33 );
1
gap> SignInt( -214378 );
-1
gap> SignInt( 0 );
0
</pre>
<p>
<a name = "SSEC001.8"></a>
<li><code>LogInt( </code><var>n</var><code>, </code><var>base</var><code> ) F</code>
<p>
<code>LogInt</code> returns the integer part of the logarithm of the positive
integer <var>n</var> with respect to the positive integer <var>base</var>, i.e., the
largest positive integer <var>exp</var> such that <i>base</i><sup><i>exp</i></sup> ≤ <i>n</i>. <code>LogInt</code>
will signal an error if either <var>n</var> or <var>base</var> is not positive.
<p>
For <var>base</var> 2 this is very efficient because the internal binary
representation of the integer is used.
<p>
<pre>
gap> LogInt( 1030, 2 );
10
gap> 2^10;
1024
gap> LogInt( 1, 10 );
0
</pre>
<p>
<a name = "SSEC001.9"></a>
<li><code>RootInt( </code><var>n</var><code> ) F</code>
<li><code>RootInt( </code><var>n</var><code>, </code><var>k</var><code> ) F</code>
<p>
<code>RootInt</code> returns the integer part of the <var>k</var>th root of the integer <var>n</var>.
If the optional integer argument <var>k</var> is not given it defaults to 2, i.e.,
<code>RootInt</code> returns the integer part of the square root in this case.
<p>
If <var>n</var> is positive, <code>RootInt</code> returns the largest positive integer <i>r</i>
such that <i>r</i><sup><i>k</i></sup> ≤ <i>n</i>. If <var>n</var> is negative and <var>k</var> is odd <code>RootInt</code>
returns <code>-RootInt( -</code><var>n</var><code>, </code><var>k</var><code> )</code>. If <var>n</var> is negative and <var>k</var> is even
<code>RootInt</code> will cause an error. <code>RootInt</code> will also cause an error if <var>k</var>
is 0 or negative.
<p>
<a name = "I2"></a>
<a name = "I2"></a>
<a name = "I3"></a>
<pre>
gap> RootInt( 361 );
19
gap> RootInt( 2 * 10^12 );
1414213
gap> RootInt( 17000, 5 );
7
gap> 7^5;
16807
</pre>
<p>
<a name = "SSEC001.10"></a>
<li><code>SmallestRootInt( </code><var>n</var><code> ) F</code>
<p>
<code>SmallestRootInt</code> returns the smallest root of the integer <var>n</var>.
<p>
The smallest root of an integer <i>n</i> is the integer <i>r</i> of smallest
absolute value for which a positive integer <i>k</i> exists such that <i>n</i> = <i>r</i><sup><i>k</i></sup>.
<p>
<a name = "I4"></a>
<pre>
gap> SmallestRootInt( 2^30 );
2
gap> SmallestRootInt( -(2^30) );
-4
</pre>
<p>
Note that (−2)<sup>30</sup> = +(2<sup>30</sup>).
<p>
<pre>
gap> SmallestRootInt( 279936 );
6
gap> LogInt( 279936, 6 );
7
gap> SmallestRootInt( 1001 );
1001
</pre>
<p>
<a name = "SSEC001.11"></a>
<li><code>Random( Integers )</code>
<p>
<code>Random</code> for integers returns
pseudo random integers between -10 and
10 distributed according to a binomial distribution.
To generate uniformly distributed integers from a range, use the
construct 'Random( [ <var>low</var> .. <var>high</var> ] )'. (Also see <a href="CHAP014.htm#SSEC005.2">Random</a>.)
<p>
<p>
<h2><a name="SECT002">14.2 Quotients and Remainders</a></h2>
<p><p>
<a name = "SSEC002.1"></a>
<li><code>QuoInt( </code><var>n</var><code>, </code><var>m</var><code> ) F</code>
<p>
<code>QuoInt</code> returns the integer part of the quotient of its integer
operands.
<p>
If <var>n</var> and <var>m</var> are positive <code>QuoInt( </code><var>n</var><code>, </code><var>m</var><code> )</code> is the largest
positive integer <var>q</var> such that <i>q</i> * <i>m</i> ≤ <i>n</i> .
If <var>n</var> or <var>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>n</var> and <var>m</var>,
and the sign of it is the product of the signs of <var>n</var> and <var>m</var>.
<p>
<code>QuoInt</code> is used in a method for the general operation
<code>EuclideanQuotient</code> (see <a href="CHAP054.htm#SSEC006.3">EuclideanQuotient</a>).
<p>
<a name = "I5"></a>
<pre>
gap> QuoInt(5,3); QuoInt(-5,3); QuoInt(5,-3); QuoInt(-5,-3);
1
-1
-1
1
</pre>
<p>
<a name = "SSEC002.2"></a>
<li><code>BestQuoInt( </code><var>n</var><code>, </code><var>m</var><code> ) F</code>
<p>
<code>BestQuoInt</code> returns the best quotient <var>q</var> of the integers <var>n</var> and <var>m</var>.
This is the quotient such that <code></code><var>n</var><code>-</code><var>q</var><code>*</code><var>m</var><code></code> 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>
<pre>
gap> BestQuoInt( 5, 3 ); BestQuoInt( -5, 3 );
2
-2
</pre>
<p>
<a name = "SSEC002.3"></a>
<li><code>RemInt( </code><var>n</var><code>, </code><var>m</var><code> ) F</code>
<p>
<code>RemInt</code> returns the remainder of its two integer operands.
<p>
If <var>m</var> is not equal to zero
<code>RemInt( </code><var>n</var><code>, </code><var>m</var><code> ) = </code><var>n</var><code> - </code><var>m</var><code> * QuoInt( </code><var>n</var><code>, </code><var>m</var><code> )</code>.
Note that the rules given for <code>QuoInt</code> imply that <code>RemInt( </code><var>n</var><code>, </code><var>m</var><code> )</code>
has the same sign as <var>n</var> and its absolute value is strictly less than the
absolute value of <var>m</var>.
Note also that <code>RemInt( </code><var>n</var><code>, </code><var>m</var><code> ) = </code><var>n</var><code> mod </code><var>m</var><code></code> when both <var>n</var> and <var>m</var>
are nonnegative.
Dividing by 0 signals an error.
<p>
<code>RemInt</code> is used in a method for the general operation
<code>EuclideanRemainder</code> (see <a href="CHAP054.htm#SSEC006.4">EuclideanRemainder</a>).
<p>
<a name = "I6"></a>
<pre>
gap> RemInt(5,3); RemInt(-5,3); RemInt(5,-3); RemInt(-5,-3);
2
-2
2
-2
</pre>
<p>
<a name = "SSEC002.4"></a>
<li><code>GcdInt( </code><var>m</var><code>, </code><var>n</var><code> ) F</code>
<p>
<code>GcdInt</code> returns the greatest common divisor of its two integer operands
<var>m</var> and <var>n</var>, i.e., the greatest integer that divides both <var>m</var> and <var>n</var>.
The greatest common divisor is never negative, even if the arguments are.
We define <code>GcdInt( </code><var>m</var><code>, 0 ) = GcdInt( 0, </code><var>m</var><code> ) = AbsInt( </code><var>m</var><code> )</code> and
<code>GcdInt( 0, 0 ) = 0</code>.
<p>
<code>GcdInt</code> is a method used by the general function <code>Gcd</code> (see <a href="CHAP054.htm#SSEC007.1">Gcd</a>).
<p>
<pre>
gap> GcdInt( 123, 66 );
3
</pre>
<p>
<a name = "SSEC002.5"></a>
<li><code>Gcdex( </code><var>m</var><code>, </code><var>n</var><code> ) F</code>
<p>
returns a record <var>g</var> describing the extended greatest common divisor of
<var>m</var> and <var>n</var>.
The component <code>gcd</code> is this gcd,
the components <code>coeff1</code> and <code>coeff2</code> are integer cofactors such that
<code></code><var>g</var><code>.gcd = </code><var>g</var><code>.coeff1 * </code><var>m</var><code> + </code><var>g</var><code>.coeff2 * </code><var>n</var><code></code>,
and the components <code>coeff3</code> and <code>coeff4</code> are integer cofactors such that
<code>0 = </code><var>g</var><code>.coeff3 * </code><var>m</var><code> + </code><var>g</var><code>.coeff4 * </code><var>n</var><code></code>.
<p>
If <var>m</var> and <var>n</var> both are nonzero, <code>AbsInt( </code><var>g</var><code>.coeff1 )</code> is less than or
equal to <code>AbsInt(</code><var>n</var><code>) / (2 * </code><var>g</var><code>.gcd)</code> and <code>AbsInt( </code><var>g</var><code>.coeff2 )</code> is less
than or equal to <code>AbsInt(</code><var>m</var><code>) / (2 * </code><var>g</var><code>.gcd)</code>.
<p>
If <var>m</var> or <var>n</var> or both are zero <code>coeff3</code> is <code>-</code><var>n</var><code> / </code><var>g</var><code>.gcd</code> and
<code>coeff4</code> is <code></code><var>m</var><code> / </code><var>g</var><code>.gcd</code>.
<p>
The coefficients always form a unimodular matrix, i.e.,
the determinant <code></code><var>g</var><code>.coeff1 * </code><var>g</var><code>.coeff4 - </code><var>g</var><code>.coeff3 * </code><var>g</var><code>.coeff2</code>
is 1 or −1.
<p>
<pre>
gap> Gcdex( 123, 66 );
rec( gcd := 3, coeff1 := 7, coeff2 := -13, coeff3 := -22, coeff4 := 41 )
</pre>
<p>
This means 3 = 7 * 123 − 13 * 66, 0 = −22 * 123 + 41 * 66.
<p>
<pre>
gap> Gcdex( 0, -3 );
rec( gcd := 3, coeff1 := 0, coeff2 := -1, coeff3 := 1, coeff4 := 0 )
gap> Gcdex( 0, 0 );
rec( gcd := 0, coeff1 := 1, coeff2 := 0, coeff3 := 0, coeff4 := 1 )
</pre>
<p>
<a name = "SSEC002.6"></a>
<li><code>LcmInt( </code><var>m</var><code>, </code><var>n</var><code> ) F</code>
<p>
returns the least common multiple of the integers <var>m</var> and <var>n</var>.
<p>
<code>LcmInt</code> is a method used by the general function <code>Lcm</code>.
<p>
<pre>
gap> LcmInt( 123, 66 );
2706
</pre>
<p>
<a name = "SSEC002.7"></a>
<li><code>CoefficientsQadic( </code><var>i</var><code>, </code><var>q</var><code> ) F</code>
<p>
returns the <var>q</var>-adic representation of the integer <var>i</var> as a list <var>l</var> of
coefficients where <i>i</i> = ∑<sub><i>j</i>=0</sub> <i>q</i><sup><i>j</i></sup> ·<i>l</i>[<i>j</i>+1].
<p>
<a name = "SSEC002.8"></a>
<li><code>CoefficientsMultiadic( </code><var>ints</var><code>, </code><var>int</var><code> ) F</code>
<p>
returns the multiadic expansion of the integer <var>int</var> modulo the integers
given in <var>ints</var> (in ascending order).
It returns a list of coefficients in the <strong>reverse</strong> order to that in <var>ints</var>.
<p>
<a name = "SSEC002.9"></a>
<li><code>ChineseRem( </code><var>moduli</var><code>, </code><var>residues</var><code> ) F</code>
<p>
<code>ChineseRem</code> returns the combination of the <var>residues</var> modulo the
<var>moduli</var>, i.e., the unique integer <var>c</var> from <code>[0..Lcm(</code><var>moduli</var><code>)-1]</code> such
that <code></code><var>c</var><code> = </code><var>residues</var><code>[i]</code> modulo <code></code><var>moduli</var><code>[i]</code> for all <var>i</var>, if it
exists. If no such combination exists <code>ChineseRem</code> signals an error.
<p>
Such a combination does exist if and only if
<code></code><var>residues</var><code>[</code><var>i</var><code>]=</code><var>residues</var><code>[</code><var>k</var><code>]</code> mod <code>Gcd(</code><var>moduli</var><code>[</code><var>i</var><code>],</code><var>moduli</var><code>[</code><var>k</var><code>])</code>
for every pair <var>i</var>, <var>k</var>. Note that this implies that such a combination
exists if the moduli are pairwise relatively prime. This is called the
Chinese remainder theorem.
<p>
<a name = "I7"></a>
<pre>
gap> ChineseRem( [ 2, 3, 5, 7 ], [ 1, 2, 3, 4 ] );
53
gap> ChineseRem( [ 6, 10, 14 ], [ 1, 3, 5 ] );
103
</pre>
<pre>
gap> ChineseRem( [ 6, 10, 14 ], [ 1, 2, 3 ] );
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
brk> gap>
</pre>
<p>
<a name = "SSEC002.10"></a>
<li><code>PowerModInt( </code><var>r</var><code>, </code><var>e</var><code>, </code><var>m</var><code> ) F</code>
<p>
returns <i>r</i><sup><i>e</i></sup> mod <i>m</i> for integers <var>r</var>,<var>e</var> and <var>m</var> (<i>e</i> ≥ 0).
Note that using <code></code><var>r</var><code> ^ </code><var>e</var><code> mod </code><var>m</var><code></code> will generally be slower,
because it can not reduce intermediate results the way <code>PowerModInt</code>
does but would compute <code></code><var>r</var><code>^</code><var>e</var><code></code> first and then reduce the result
afterwards.
<p>
<code>PowerModInt</code> is a method for the general operation <code>PowerMod</code>.
<p>
<p>
<h2><a name="SECT003">14.3 Prime Integers and Factorization</a></h2>
<p><p>
<a name = "SSEC003.1"></a>
<li><code>Primes V</code>
<p>
<code>Primes</code> is a strictly sorted list of the 168 primes less than 1000.
<p>
This is used in <code>IsPrimeInt</code> and <code>FactorsInt</code> to cast out small primes
quickly.
<p>
<pre>
gap> Primes[1];
2
gap> Primes[100];
541
</pre>
<p>
<a name = "SSEC003.2"></a>
<li><code>IsPrimeInt( </code><var>n</var><code> ) F</code>
<a name = "SSEC003.2"></a>
<li><code>IsProbablyPrimeInt( </code><var>n</var><code> ) F</code>
<p>
<code>IsPrimeInt</code> returns <code>false</code> if it can prove that <var>n</var> is composite and
<code>true</code> otherwise.
By convention <code>IsPrimeInt(0) = IsPrimeInt(1) = false</code>
and we define <code>IsPrimeInt( -</code><var>n</var><code> ) = IsPrimeInt( </code><var>n</var><code> )</code>.
<p>
<code>IsPrimeInt</code> will return <code>true</code> for every prime <i>n</i>. <code>IsPrimeInt</code> will
return <code>false</code> for all composite <i>n</i> < 10<sup>13</sup> and for all composite <i>n</i>
that have a factor <i>p</i> < 1000. So for integers <i>n</i> < 10<sup>13</sup>,
<code>IsPrimeInt</code> is a proper primality test. It is conceivable that
<code>IsPrimeInt</code> may return <code>true</code> for some composite <i>n</i> > 10<sup>13</sup>, but no
such <i>n</i> is currently known. So for integers <i>n</i> > 10<sup>13</sup>, <code>IsPrimeInt</code>
is a probable-primality test. <code>IsPrimeInt</code> will issue a
warning when its argument is probably prime but not a proven prime.
(The function <code>IsProbablyPrimeInt</code> will do the same calculations but not
issue a warning.) The warning can be switched off by
<code>SetInfoLevel( InfoPrimeInt, 0 );</code>, the default level is 1.
<p>
If composites that fool <code>IsPrimeInt</code> do exist, they would be extremely
rare, and finding one by pure chance might be less likely than finding a
bug in <font face="Gill Sans,Helvetica,Arial">GAP</font>. We would appreciate being informed about any example of a
composite number <var>n</var> for which <code>IsPrimeInt</code> returns <code>true</code>.
<p>
<code>IsPrimeInt</code> is a deterministic algorithm, i.e., the computations involve
no random numbers, and repeated calls will always return the same result.
<code>IsPrimeInt</code> first does trial divisions by the primes less than 1000.
Then it tests that <i>n</i> is a strong pseudoprime w.r.t. the base 2.
Finally it tests whether <i>n</i> is a Lucas pseudoprime w.r.t. the smallest
quadratic nonresidue of <i>n</i>. A better description can be found in the
comment in the library file <code>integer.gi</code>.
<p>
The time taken by <code>IsPrimeInt</code> is approximately proportional to the third
power of the number of digits of <var>n</var>. Testing numbers with several
hundreds digits is quite feasible.
<p>
<code>IsPrimeInt</code> is a method for the general operation <code>IsPrime</code>.
<p>
Remark: In future versions of <font face="Gill Sans,Helvetica,Arial">GAP</font> we hope to change the definition of
<code>IsPrimeInt</code> to return <code>true</code> only for proven primes (currently, we lack
a sufficiently good primality proving function). In applications, use
explicitly <code>IsPrimeInt</code> or <code>IsProbablePrimeInt</code> with this change in
mind.
<p>
<pre>
gap> IsPrimeInt( 2^31 - 1 );
true
gap> IsPrimeInt( 10^42 + 1 );
false
</pre>
<p>
<a name = "SSEC003.3"></a>
<li><code>IsPrimePowerInt( </code><var>n</var><code> ) F</code>
<p>
<code>IsPrimePowerInt</code> returns <code>true</code> if the integer <var>n</var> is a prime power and
<code>false</code> otherwise.
<p>
<i>n</i> is a <strong>prime power</strong> if there exists a prime <i>p</i> and a positive integer
<i>i</i> such that <i>p</i><sup><i>i</i></sup> = <i>n</i>. If <i>n</i> is negative the condition is that there
must exist a negative prime <i>p</i> and an odd positive integer <i>i</i> such that
<i>p</i><sup><i>i</i></sup> = <i>n</i>. 1 and -1 are not prime powers.
<p>
Note that <code>IsPrimePowerInt</code> uses <code>SmallestRootInt</code> (see
<a href="CHAP014.htm#SSEC001.10">SmallestRootInt</a>) and a probable-primality test (see <a href="CHAP014.htm#SSEC003.2">IsPrimeInt</a>).
<p>
<pre>
gap> IsPrimePowerInt( 31^5 );
true
gap> IsPrimePowerInt( 2^31-1 ); # 2^31-1 is actually a prime
true
gap> IsPrimePowerInt( 2^63-1 );
false
gap> Filtered( [-10..10], IsPrimePowerInt );
[ -8, -7, -5, -3, -2, 2, 3, 4, 5, 7, 8, 9 ]
</pre>
<p>
<a name = "SSEC003.4"></a>
<li><code>NextPrimeInt( </code><var>n</var><code> ) F</code>
<p>
<code>NextPrimeInt</code> returns the smallest prime which is strictly larger than
the integer <var>n</var>.
<p>
Note that <code>NextPrimeInt</code> uses a probable-primality test (see
<a href="CHAP014.htm#SSEC003.2">IsPrimeInt</a>).
<p>
<pre>
gap> NextPrimeInt( 541 ); NextPrimeInt( -1 );
547
2
</pre>
<p>
<a name = "SSEC003.5"></a>
<li><code>PrevPrimeInt( </code><var>n</var><code> ) F</code>
<p>
<code>PrevPrimeInt</code> returns the largest prime which is strictly smaller than
the integer <var>n</var>.
<p>
Note that <code>PrevPrimeInt</code> uses a probable-primality test (see
<a href="CHAP014.htm#SSEC003.2">IsPrimeInt</a>).
<p>
<pre>
gap> PrevPrimeInt( 541 ); PrevPrimeInt( 1 );
523
-2
</pre>
<p>
<a name = "SSEC003.6"></a>
<li><code>FactorsInt( </code><var>n</var><code> ) F</code>
<li><code>FactorsInt( </code><var>n</var><code> : RhoTrials := </code><var>trials</var><code> ) F</code>
<p>
<code>FactorsInt</code> returns a list of prime factors of the integer <var>n</var>.
<p>
If the <var>i</var>th power of a prime divides <var>n</var> this prime appears <var>i</var> times.
The list is sorted, that is the smallest prime factors come first.
The first element has the same sign as <var>n</var>, the others are positive.
For any integer <var>n</var> it holds that <code>Product( FactorsInt( </code><var>n</var><code> ) ) = </code><var>n</var><code></code>.
<p>
Note that <code>FactorsInt</code> uses a probable-primality test (see <a href="CHAP014.htm#SSEC003.2">IsPrimeInt</a>).
Thus <code>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>SetInfoLevel(InfoPrimeInt, 0);</code>.
<p>
The time taken by <code>FactorsInt</code> is approximately proportional to the
square root of the second largest prime factor of <var>n</var>, which is the last
one that <code>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>n</var>. <code>FactorsInt</code> is guaranteed to find
all factors less than 10<sup>6</sup> and will find most factors less than
10<sup>10</sup>. If <var>n</var> contains multiple factors larger than that
<code>FactorsInt</code> may not be able to factor <var>n</var> and will then signal an error.
<p>
<code>FactorsInt</code> is used in a method for the general operation <code>Factors</code>.
<p>
In the second form, FactorsInt calls FactorsRho with a limit of <var>trials</var>
on the number of trials is performs. The default is 8192.
<p>
<pre>
gap> FactorsInt( -Factorial(6) );
[ -2, 2, 2, 2, 3, 3, 5 ]
gap> Set( FactorsInt( Factorial(13)/11 ) );
[ 2, 3, 5, 7, 13 ]
gap> FactorsInt( 2^63 - 1 );
[ 7, 7, 73, 127, 337, 92737, 649657 ]
gap> FactorsInt( 10^42 + 1 );
#I IsPrimeInt: probably prime, but not proven: 4458192223320340849
[ 29, 101, 281, 9901, 226549, 121499449, 4458192223320340849 ]
</pre>
<p>
<a name = "SSEC003.7"></a>
<li><code>PartialFactorization( </code><var>n</var><code> ) O</code>
<li><code>PartialFactorization( </code><var>n</var><code>, </code><var>effort</var><code> ) O</code>
<p>
<code>PartialFactorization</code> returns a partial factorization of the integer <var>n</var>.
No assertions are made about the primality of the factors, except of
those mentioned below.
<p>
The argument <var>effort</var>, if given, specifies how intensively the function
should try to determine factors of <var>n</var>. The default is <var>effort</var> = 5.
<p>
<dl compact>
<dt>-<dd>If <var>effort</var> = 0, trial division by the primes below 100 is
done. Returned factors below 10<sup>4</sup> are guaranteed to be
prime.
<dt>-<dd>If <var>effort</var> = 1, trial division by the primes below 1000 is
done. Returned factors below 10<sup>6</sup> are guaranteed to be
prime.
<dt>-<dd>If <var>effort</var> = 2, additionally trial division by the numbers
in the lists <code>Primes2</code> and <code>ProbablePrimes2</code> is done, and
perfect powers are detected. Returned factors below 10<sup>6</sup>
are guaranteed to be prime.
<dt>-<dd>If <var>effort</var> = 3, additionally <code>FactorsRho</code> (Pollard's Rho)
with <var>RhoTrials</var> = 256 is used.
<dt>-<dd>If <var>effort</var> = 4, as above, but <var>RhoTrials</var> = 2048.
<dt>-<dd>If <var>effort</var> = 5, as above, but <var>RhoTrials</var> = 8192.
Returned factors below 10<sup>12</sup> are guaranteed to be prime,
and all prime factors below 10<sup>6</sup> are guaranteed to be found.
<dt>-<dd>If <var>effort</var> = 6 and <font face="Gill Sans,Helvetica,Arial">FactInt</font> is loaded, in addition to
the above quite a number of special cases are handled.
<dt>-<dd>If <var>effort</var> = 7 and <font face="Gill Sans,Helvetica,Arial">FactInt</font> 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.
</dl>
Increasing the value of the argument <var>effort</var> by one usually results
in an increase of the runtime requirements by a factor of (very roughly!)
3 to 10.
<p>
<a name = "I8"></a>
<pre>
gap> List([0..5],i->PartialFactorization(7^64-1,i));
[ [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17,
1868505648951954197516197706132003401892793036353 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17, 353,
5293217135841230021292344776577913319809612001 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17, 353, 134818753, 47072139617,
531968664833, 1567903802863297 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17, 353, 1201, 169553, 7699649,
134818753, 47072139617, 531968664833 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17, 353, 1201, 169553, 7699649,
134818753, 47072139617, 531968664833 ],
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 5, 17, 353, 1201, 169553, 7699649,
134818753, 47072139617, 531968664833 ] ]
</pre>
<p>
<a name = "SSEC003.8"></a>
<li><code>PrintFactorsInt( </code><var>n</var><code> ) F</code>
<p>
prints the prime factorization of the integer <var>n</var> in human-readable
form.
<p>
<pre>
gap> PrintFactorsInt( Factorial( 7 ) ); Print( "\n" );
2^4*3^2*5*7
</pre>
<p>
<a name = "SSEC003.9"></a>
<li><code>PrimePowersInt( </code><var>n</var><code> ) F</code>
<p>
returns the prime factorization of the integer <var>n</var> as a list
[ <i>p</i><sub>1</sub>, <i>e</i><sub>1</sub>, …, <i>p</i><sub><i>n</i></sub>, <i>e</i><sub><i>n</i></sub> ] with <i>n</i> = ∏<sub><i>i</i>=1</sub><sup><i>n</i></sup> <i>p</i><sub><i>i</i></sub><sup><i>e</i><sub><i>i</i></sub></sup>.
<p>
<pre>
gap> PrimePowersInt( Factorial( 7 ) );
[ 2, 4, 3, 2, 5, 1, 7, 1 ]
</pre>
<p>
<a name = "SSEC003.10"></a>
<li><code>DivisorsInt( </code><var>n</var><code> ) F</code>
<p>
<code>DivisorsInt</code> returns a list of all divisors of the integer <var>n</var>. The
list is sorted, so that it starts with 1 and ends with <var>n</var>. We define
that <code>Divisors( -</code><var>n</var><code> ) = Divisors( </code><var>n</var><code> )</code>.
<p>
Since the set of divisors of 0 is infinite calling <code>DivisorsInt( 0 )</code>
causes an error.
<p>
<code>DivisorsInt</code> may call <code>FactorsInt</code> to obtain the prime factors.
<code>Sigma</code> and <code>Tau</code> (see <a href="CHAP015.htm#SSEC004.1">Sigma</a> and <a href="CHAP015.htm#SSEC004.2">Tau</a>) compute the sum and the
number of positive divisors, respectively.
<p>
<a name = "I9"></a>
<pre>
gap> DivisorsInt( 1 ); DivisorsInt( 20 ); DivisorsInt( 541 );
[ 1 ]
[ 1, 2, 4, 5, 10, 20 ]
[ 1, 541 ]
</pre>
<p>
<p>
<h2><a name="SECT004">14.4 Residue Class Rings</a></h2>
<p><p>
<a name = "I10"></a>
<a name = "SSEC004.1"></a>
<li><code></code><var>r</var><code> / </code><var>s</var><code> mod </code><var>n</var><code></code>
<p>
If <var>r</var>, <var>s</var> and <var>n</var> are integers, <code></code><var>r</var><code> / </code><var>s</var><code></code> as a reduced fraction is
<code></code><var>p</var><code> / </code><var>q</var><code></code>, and <var>q</var> and <var>n</var> are coprime, then <code></code><var>r</var><code> / </code><var>s</var><code> mod </code><var>n</var><code></code> is
defined to be the product of <var>p</var> and the inverse of <var>q</var> modulo <var>n</var>. See
Section <a href="CHAP004.htm#SECT012">Arithmetic Operators</a> for more details and definitions.
<p>
With the above definition, <code>4 / 6 mod 32</code> equals <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>
<a name = "SSEC004.2"></a>
<li><code>ZmodnZ( </code><var>n</var><code> ) F</code>
<a name = "SSEC004.2"></a>
<li><code>ZmodpZ( </code><var>p</var><code> ) F</code>
<a name = "SSEC004.2"></a>
<li><code>ZmodpZNC( </code><var>p</var><code> ) F</code>
<p>
<code>ZmodnZ</code> returns a ring <i>R</i> isomorphic to the residue class ring of the
integers modulo the positive integer <var>n</var>.
The element corresponding to the residue class of the integer <i>i</i> in this
ring can be obtained by <i>i</i> * <tt>One</tt>( <i>R</i> ), and a representative of the
residue class corresponding to the element <i>x</i> ∈ <i>R</i> can be computed by
<tt>Int</tt>( <i>x</i> ).
<p>
<a name = "I11"></a>
<code>ZmodnZ( </code><var>n</var><code> )</code> is equivalent to <code>Integers mod </code><var>n</var><code></code>.
<p>
<code>ZmodpZ</code> does the same if the argument <var>p</var> is a prime integer,
additionally the result is a field.
<code>ZmodpZNC</code> omits the check whether <var>p</var> is a prime.
<p>
Each ring returned by these functions contains the whole family of its
elements
if <i>n</i> is not a prime, and is embedded into the family of finite field
elements of characteristic <i>n</i> if <i>n</i> is a prime.
<p>
<a name = "SSEC004.3"></a>
<li><code>ZmodnZObj( </code><var>Fam</var><code>, </code><var>r</var><code> ) O</code>
<li><code>ZmodnZObj( </code><var>r</var><code>, </code><var>n</var><code> ) O</code>
<p>
If the first argument is a residue class family <var>Fam</var> then <code>ZmodnZObj</code>
returns the element in <var>Fam</var> whose coset is represented by the integer
<var>r</var>.
If the two arguments are an integer <var>r</var> and a positive integer <var>n</var> then
<code>ZmodnZObj</code> returns the element in <code>ZmodnZ( </code><var>n</var><code> )</code> (see <a href="CHAP014.htm#SSEC004.2">ZmodnZ</a>)
whose coset is represented by the integer <var>r</var>.
<p>
<pre>
gap> r:= ZmodnZ(15);
(Integers mod 15)
gap> fam:=ElementsFamily(FamilyObj(r));;
gap> a:= ZmodnZObj(fam,9);
ZmodnZObj( 9, 15 )
gap> a+a;
ZmodnZObj( 3, 15 )
gap> Int(a+a);
3
</pre>
<p>
<a name = "SSEC004.4"></a>
<li><code>IsZmodnZObj( </code><var>obj</var><code> ) C</code>
<a name = "SSEC004.4"></a>
<li><code>IsZmodnZObjNonprime( </code><var>obj</var><code> ) C</code>
<a name = "SSEC004.4"></a>
<li><code>IsZmodpZObj( </code><var>obj</var><code> ) C</code>
<a name = "SSEC004.4"></a>
<li><code>IsZmodpZObjSmall( </code><var>obj</var><code> ) C</code>
<a name = "SSEC004.4"></a>
<li><code>IsZmodpZObjLarge( </code><var>obj</var><code> ) C</code>
<p>
The elements in the rings <i>Z</i> / <i>n</i> <i>Z</i> are in the category <code>IsZmodnZObj</code>.
If <i>n</i> is a prime then the elements are of course also in the category
<code>IsFFE</code> (see <a href="CHAP057.htm#SSEC001.1">IsFFE</a>), otherwise they are in <code>IsZmodnZObjNonprime</code>.
<code>IsZmodpZObj</code> is an abbreviation of <code>IsZmodnZObj and IsFFE</code>. This
category is the disjoint union of <code>IsZmodpZObjSmall</code> and
<code>IsZmodpZObjLarge</code>, the former containing all elements with <i>n</i> at most
<code>MAXSIZE_GF_INTERNAL</code>.
<p>
The reasons to distinguish the prime case from the nonprime case are
<ul>
<li>
that objects in <code>IsZmodnZObjNonprime</code> have an external representation
(namely the residue in the range [ 0, 1, …, <i>n</i>−1 ]),
<li>
that the comparison of elements can be defined as comparison of the
residues, and
<li>
that the elements lie in a family of type <code>IsZmodnZObjNonprimeFamily</code>
(note that for prime <i>n</i>, the family must be an <code>IsFFEFamily</code>).
</ul>
<p>
The reasons to distinguish the small and the large case are
that for small <i>n</i> 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 <i>n</i>.
<p>
Note that we <strong>cannot</strong> claim that every finite field element of degree 1
is in <code>IsZmodnZObj</code>, since finite field elements in internal
representation may not know that they lie in the prime field.
<p>
The residue class rings are rings, thus all operations for rings (see
Chapter <a href="CHAP054.htm">Rings</a>) apply.
See also Chapters <a href="CHAP057.htm">Finite fields</a> and <a href="CHAP015.htm">Number theory</a>.
<p>
<p>
<h2><a name="SECT005">14.5 Random Sources</a></h2>
<p><p>
<font face="Gill Sans,Helvetica,Arial">GAP</font> provides <code>Random</code> methods (see <a href="CHAP014.htm#SSEC005.2">Random</a>) for many collections of
objects.
On a lower level these methods use <strong>random sources</strong> which provide
random integers and random choices from lists.
<p>
<a name = "SSEC005.1"></a>
<li><code>IsRandomSource( </code><var>rs</var><code> ) C</code>
<p>
This is the category of random source objects <var>rs</var> which are defined to
have methods available for the following operations which are explained
in more detail below: <code>Random( </code><var>rs</var><code>, </code><var>list</var><code> )</code> giving a random element
of a list, <code>Random( </code><var>rs</var><code>, </code><var>low</var><code>, </code><var>high</var><code> )</code> giving a random integer between
<var>low</var> and <var>high</var> (inclusive), <code>Init</code>, <code>State</code> and <code>Reset</code>.
<p>
Use <code>RandomSource</code> (see <a href="CHAP014.htm#SSEC005.5">RandomSource</a>) to construct new random sources.
<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>
Random source objects lie in the family <code>RandomSourcesFamily</code>.
<p>
<a name = "SSEC005.2"></a>
<li><code>Random( </code><var>rs</var><code>, </code><var>list</var><code> ) O</code>
<li><code>Random( </code><var>rs</var><code>, </code><var>low</var><code>, </code><var>high</var><code> ) O</code>
<p>
This operation returns a random element from list <var>list</var>, or an integer
in the range from the given (possibly large) integers <var>low</var> to <var>high</var>,
respectively.
The choice should only depend on the random source <var>rs</var> and have no
effect on other random sources.
<p>
<a name = "SSEC005.3"></a>
<li><code>State( </code><var>rs</var><code> ) O</code>
<a name = "SSEC005.3"></a>
<li><code>Reset( </code><var>rs</var><code> ) O</code>
<li><code>Reset( </code><var>rs</var><code>, </code><var>seed</var><code> ) O</code>
<a name = "SSEC005.3"></a>
<li><code>Init( </code><var>rs</var><code> ) O</code>
<li><code>Init( </code><var>prers</var><code>, </code><var>seed</var><code> ) O</code>
<p>
These are the basic operations for which random sources (see
<a href="CHAP014.htm#SSEC005.1">IsRandomSource</a>) must have methods.
<p>
<code>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 truely random physical data) then <code>State</code> should return
<code>fail</code>.
<p>
<code>Reset( </code><var>rs</var><code>, </code><var>seed</var><code> )</code> resets the random source <var>rs</var> to a state described
by <var>seed</var>, if the random source can be reset (otherwise it should do
nothing). Here <var>seed</var> can be an output of <code>State</code> and then should reset
to that state. Also, the methods should always allow integers as <var>seed</var>.
Without the <var>seed</var> argument the default <i>seed</i> = 1 is used.
<p>
<code>Init</code> is the constructor of a random source, it gets an empty component
object 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>seed</var> for the initial state, as explained for <code>Reset</code>.
<p>
<a name = "SSEC005.4"></a>
<li><code>IsGlobalRandomSource( </code><var>rs</var><code> ) C</code>
<a name = "SSEC005.4"></a>
<li><code>IsGAPRandomSource( </code><var>rs</var><code> ) C</code>
<a name = "SSEC005.4"></a>
<li><code>IsMersenneTwister( </code><var>rs</var><code> ) C</code>
<a name = "SSEC005.4"></a>
<li><code>GlobalRandomSource V</code>
<a name = "SSEC005.4"></a>
<li><code>GlobalMersenneTwister V</code>
<p>
Currently, the <font face="Gill Sans,Helvetica,Arial">GAP</font> library provides three types of random sources,
distinguished by the three listed categories.
<p>
<code>IsGlobalRandomSource</code> gives access to the <strong>classical</strong> global
random generator which was used by <font face="Gill Sans,Helvetica,Arial">GAP</font> in previous 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>GlobalRandomSource</code>. This uses the additive random number
generator described in <a href="biblio.htm#TACP2"><cite>TACP2</cite></a> (Algorithm A in 3.2.2 with lag 30).
<p>
<code>IsGAPRandomSource</code> uses the same number generator as
<code>IsGlobalRandomSource</code>, but you can create several of these random sources
which generate their random numbers independently of all other random
sources.
<p>
<code>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 2<sup>19937</sup>−1 and the numbers have a 623-dimensional
equidistribution. For more details and the origin of the code used in the
<font face="Gill Sans,Helvetica,Arial">GAP</font> kernel, see:
<p>
<code>http://www.math.sci.hiroshima-u.ac.jp/ m-mat/MT/emt.html</code>
<p>
Use the Mersenne twister if possible, in particular for generating many
large random integers.
<p>
There is also a predefined global random source <code>GlobalMersenneTwister</code>.
<p>
<a name = "SSEC005.5"></a>
<li><code>RandomSource( </code><var>cat</var><code> ) O</code>
<li><code>RandomSource( </code><var>cat</var><code>, </code><var>seed</var><code> ) O</code>
<p>
This operation is used to create new random sources. The first argument is
the category describing the type of the random generator, an optional
<var>seed</var> which can be an integer or a type specific data structure can be
given to specify the initial state.
<p>
<pre>
gap> rs1 := RandomSource(IsMersenneTwister);
<RandomSource in IsMersenneTwister>
gap> state1 := State(rs1);;
gap> l1 := List([1..10000], i-> Random(rs1, [1..6]));;
gap> rs2 := RandomSource(IsMersenneTwister);;
gap> l2 := List([1..10000], i-> Random(rs2, [1..6]));;
gap> l1 = l2;
true
gap> l1 = List([1..10000], i-> Random(rs1, [1..6]));
false
gap> n := Random(rs1, 1, 2^220);
1598617776705343302477918831699169150767442847525442557699717518961
</pre>
<p>
<p>
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<font face="Gill Sans,Helvetica,Arial">GAP 4 manual<br>December 2008
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