SSL numtheor.gd Sprache: unbekannt

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Martin Schönert.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file declares operations for integer primes.

## <#GAPDoc Label="[1]{numtheor}">
## &GAP; provides a couple of elementary number theoretic functions.
## Most of these deal with the group of integers coprime to <M>m</M>,
## called the <E>prime residue group</E>.
## The order of this group is <M>\phi(m)</M> (see <Ref Oper="Phi"/>),
## and <M>\lambda(m)</M> (see <Ref Oper="Lambda"/>) is its exponent.
## This group is cyclic if and only if <M>m</M> is 2, 4,
## an odd prime power <M>p^n</M>, or twice an odd prime power <M>2 p^n</M>.
## In this case the generators of the group, i.e., elements of order
## <M>\phi(m)</M>,
## are called <E>primitive roots</E>
## (see <Ref Func="PrimitiveRootMod"/>).
## 
## Note that neither the arguments nor the return values of the functions
## listed below are groups or group elements in the sense of &GAP;.
## The arguments are simply integers.
## <#/GAPDoc>
##

##########################################################################
##
#V InfoNumtheor
##
## <#GAPDoc Label="InfoNumtheor">
## <ManSection>
## <InfoClass Name="InfoNumtheor"/>
##
## <Description>
## <Ref InfoClass="InfoNumtheor"/> is the info class
## (see <Ref Sect="Info Functions"/>)
## for the functions in the number theory chapter.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareInfoClass( "InfoNumtheor" );

#############################################################################
##
#F PrimeResidues( <m> ) . . . . . . . integers relative prime to an integer
##
## <#GAPDoc Label="PrimeResidues">
## <ManSection>
## <Func Name="PrimeResidues" Arg='m'/>
##
## <Description>
## <Ref Func="PrimeResidues"/> returns the set of integers from the range
## <C>[ 0 .. Abs( <A>m</A> )-1 ]</C>
## that are coprime to the integer <A>m</A>.
## 
## <C>Abs(<A>m</A>)</C> must be less than <M>2^{28}</M>,
## otherwise the set would probably be too large anyhow.
## 
## <Example><![CDATA[
## gap> PrimeResidues( 0 ); PrimeResidues( 1 ); PrimeResidues( 20 );
## [ ]
## [ 0 ]
## [ 1, 3, 7, 9, 11, 13, 17, 19 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PrimeResidues" );

#############################################################################
##
#O Phi( <m> ) . . . . . . . . . . . . . . . . . . Euler's totient function
##
## <#GAPDoc Label="Phi">
## <ManSection>
## <Oper Name="Phi" Arg='m'/>
##
## <Description>
## <Index Subkey="of the prime residue group">order</Index>
## <Index Subkey="order">prime residue group</Index>
## <Index>Euler's totient function</Index>
## <Ref Oper="Phi"/> returns the number <M>\phi(<A>m</A>)</M> of positive
## integers less than the positive integer <A>m</A>
## that are coprime to <A>m</A>.
## 
## Suppose that <M>m = p_1^{{e_1}} p_2^{{e_2}} \cdots p_k^{{e_k}}</M>.
## Then <M>\phi(m)</M> is
## <M>p_1^{{e_1-1}} (p_1-1) p_2^{{e_2-1}} (p_2-1) \cdots p_k^{{e_k-1}} (p_k-1)</M>.
## <Example><![CDATA[
## gap> Phi( 12 );
## 4
## gap> Phi( 2^13-1 ); # this proves that 2^(13)-1 is a prime
## 8190
## gap> Phi( 2^15-1 );
## 27000
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Phi", [ IsObject ] );

#############################################################################
##
#O Lambda( <m> ) . . . . . . . . . . . . . . . . . . . . Carmichael function
##
## <#GAPDoc Label="Lambda">
## <ManSection>
## <Oper Name="Lambda" Arg='m'/>
##
## <Description>
## <Index>Carmichael's lambda function</Index>
## <Index Subkey="exponent">prime residue group</Index>
## <Index Subkey="of the prime residue group">exponent</Index>
## <Ref Oper="Lambda"/> returns the exponent <M>\lambda(<A>m</A>)</M>
## of the group of prime residues modulo the integer <A>m</A>.
## 
## <M>\lambda(<A>m</A>)</M> is the smallest positive integer <M>l</M> such that for every
## <M>a</M> relatively prime to <A>m</A> we have <M>a^l \equiv 1 \pmod{<A>m</A>}</M>.
## Fermat's theorem asserts
## <M>a^{{\phi(<A>m</A>)}} \equiv 1 \pmod{<A>m</A>}</M>;
## thus <M>\lambda(<A>m</A>)</M> divides <M>\phi(<A>m</A>)</M> (see <Ref Oper="Phi"/>).
## 
## Carmichael's theorem states that <M>\lambda</M> can be computed as follows:
## <M>\lambda(2) = 1</M>, <M>\lambda(4) = 2</M> and
## <M>\lambda(2^e) = 2^{{e-2}}</M>
## if <M>3 \leq e</M>,
## <M>\lambda(p^e) = (p-1) p^{{e-1}}</M> (i.e. <M>\phi(m)</M>) if <M>p</M>
## is an odd prime and
## <M>\lambda(m*n) = </M><C>Lcm</C><M>( \lambda(m), \lambda(n) )</M> if <M>m, n</M> are coprime.
## 
## Composites for which <M>\lambda(m)</M> divides <M>m - 1</M> are called Carmichaels.
## If <M>6k+1</M>, <M>12k+1</M> and <M>18k+1</M> are primes their product is such a number.
## There are only 1547 Carmichaels below <M>10^{10}</M> but 455052511 primes.
## <Example><![CDATA[
## gap> Lambda( 10 );
## 4
## gap> Lambda( 30 );
## 4
## gap> Lambda( 561 ); # 561 is the smallest Carmichael number
## 80
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Lambda", [ IsObject ] );

#############################################################################
##
#F OrderMod( <n>, <m>[, <bound>] ) . . . multiplicative order of an integer
##
## <#GAPDoc Label="OrderMod">
## <ManSection>
## <Func Name="OrderMod" Arg='n, m[, bound]'/>
##
## <Description>
## <Index>multiplicative order of an integer</Index>
## <Ref Func="OrderMod"/> returns the multiplicative order of the integer
## <A>n</A> modulo the positive integer <A>m</A>.
## If <A>n</A> and <A>m</A> are not coprime the order of <A>n</A> is not
## defined and <Ref Func="OrderMod"/> will return <C>0</C>.
## 
## If <A>n</A> and <A>m</A> are relatively prime the multiplicative order of
## <A>n</A> modulo <A>m</A> is the smallest positive integer <M>i</M>
## such that <M><A>n</A>^i \equiv 1 \pmod{<A>m</A>}</M>.
## If the group of prime residues modulo <A>m</A> is cyclic then
## each element of maximal order is called a primitive root modulo <A>m</A>
## (see <Ref Func="IsPrimitiveRootMod"/>).
## 
## If no a priori known multiple <A>bound</A> of the desired order is given,
## <Ref Func="OrderMod"/> usually spends most of its time factoring <A>m</A>
## for computing <M>\lambda(<A>m</A>)</M> (see <Ref Oper="Lambda"/>) as the
## default for <A>bound</A>, and then factoring <A>bound</A>
## (see <Ref Func="FactorsInt"/>).
## 
## If an incorrect <A>bound</A> is given then the result will be wrong.
## <Example><![CDATA[
## gap> OrderMod( 2, 7 );
## 3
## gap> OrderMod( 3, 7 ); # 3 is a primitive root modulo 7
## 6
## gap> m:= (5^166-1) / 167;; # about 10^113
## gap> OrderMod( 5, m, 166 ); # needs minutes without third argument
## 166
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "OrderMod" );

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##
#F IsPrimitiveRootMod( <r>, <m> ) . . . . . . . . test for a primitive root
##
## <#GAPDoc Label="IsPrimitiveRootMod">
## <ManSection>
## <Func Name="IsPrimitiveRootMod" Arg='r, m'/>
##
## <Description>
## <Index Subkey="for a primitive root">test</Index>
## <Index Subkey="generator">prime residue group</Index>
## <Index Subkey="of the prime residue group">generator</Index>
## <Ref Func="IsPrimitiveRootMod"/> returns <K>true</K> if the integer
## <A>r</A> is a primitive root modulo the positive integer <A>m</A>,
## and <K>false</K> otherwise.
## If <A>r</A> is less than 0 or larger than <A>m</A> it is replaced by its
## remainder.
## <Example><![CDATA[
## gap> IsPrimitiveRootMod( 2, 541 );
## true
## gap> IsPrimitiveRootMod( -539, 541 ); # same computation as above;
## true
## gap> IsPrimitiveRootMod( 4, 541 );
## false
## gap> ForAny( [1..29], r -> IsPrimitiveRootMod( r, 30 ) );
## false
## gap> # there is no a primitive root modulo 30
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IsPrimitiveRootMod" );

#############################################################################
##
#F PrimitiveRootMod( <m>[, <start>] ) . . primitive root modulo an integer
##
## <#GAPDoc Label="PrimitiveRootMod">
## <ManSection>
## <Func Name="PrimitiveRootMod" Arg='m[, start]'/>
##
## <Description>
## <Index>primitive root modulo an integer</Index>
## <Index Subkey="generator">prime residue group</Index>
## <Index Subkey="of the prime residue group">generator</Index>
## <Ref Func="PrimitiveRootMod"/> returns the smallest primitive root modulo
## the positive integer <A>m</A> and <K>fail</K> if no such primitive root
## exists.
## If the optional second integer argument <A>start</A> is given
## <Ref Func="PrimitiveRootMod"/> returns the smallest primitive root that
## is strictly larger than <A>start</A>.
## <Example><![CDATA[
## gap> # largest primitive root for a prime less than 2000:
## gap> PrimitiveRootMod( 409 );
## 21
## gap> PrimitiveRootMod( 541, 2 );
## 10
## gap> # 327 is the largest primitive root mod 337:
## gap> PrimitiveRootMod( 337, 327 );
## fail
## gap> # there exists no primitive root modulo 30:
## gap> PrimitiveRootMod( 30 );
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PrimitiveRootMod" );

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##
#F GeneratorsPrimeResidues( <n> ) . . . . . . generators of the Galois group
##
## <#GAPDoc Label="GeneratorsPrimeResidues">
## <ManSection>
## <Func Name="GeneratorsPrimeResidues" Arg='n'/>
##
## <Description>
## Let <A>n</A> be a positive integer.
## <Ref Func="GeneratorsPrimeResidues"/> returns a description of generators
## of the group of prime residues modulo <A>n</A>.
## The return value is a record with components
## <List>
## <C>primes</C>: 
## <Item>
## a list of the prime factors of <A>n</A>,
## </Item>
## <C>exponents</C>: 
## <Item>
## a list of the exponents of these primes in the factorization of <A>n</A>,
## and
## </Item>
## <C>generators</C>: 
## <Item>
## a list describing generators of the group of prime residues;
## for the prime factor <M>2</M>, either a primitive root or a list of two
## generators is stored,
## for each other prime factor of <A>n</A>, a primitive root is stored.
## </Item>
## </List>
## <Example><![CDATA[
## gap> GeneratorsPrimeResidues( 1 );
## rec( exponents := [ ], generators := [ ], primes := [ ] )
## gap> GeneratorsPrimeResidues( 4*3 );
## rec( exponents := [ 2, 1 ], generators := [ 7, 5 ],
## primes := [ 2, 3 ] )
## gap> GeneratorsPrimeResidues( 8*9*5 );
## rec( exponents := [ 3, 2, 1 ],
## generators := [ [ 271, 181 ], 281, 217 ], primes := [ 2, 3, 5 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GeneratorsPrimeResidues" );

#############################################################################
##
#F Jacobi( <n>, <m> ) . . . . . . . . . . . . . . . . . . . . Jacobi symbol
##
## <#GAPDoc Label="Jacobi">
## <ManSection>
## <Func Name="Jacobi" Arg='n, m'/>
##
## <Description>
## <Index>quadratic residue</Index>
## <Index Subkey="quadratic">residue</Index>
## <Ref Func="Jacobi"/> returns the value of the
## <E>Kronecker-Jacobi symbol</E> <M>J(<A>n</A>,<A>m</A>)</M> of the integer
## <A>n</A> modulo the integer <A>m</A>.
## It is defined as follows:
## 
## If <M>n</M> and <M>m</M> are not coprime then <M>J(n,m) = 0</M>.
## Furthermore, <M>J(n,1) = 1</M> and <M>J(n,-1) = -1</M> if <M>m < 0</M>
## and <M>+1</M> otherwise.
## And for odd <M>n</M> it is <M>J(n,2) = (-1)^k</M> with
## <M>k = (n^2-1)/8</M>.
## For odd primes <M>m</M> which are coprime to <M>n</M> the
## Kronecker-Jacobi symbol has the same value as the Legendre symbol
## (see <Ref Func="Legendre"/>).
## 
## For the general case suppose that <M>m = p_1 \cdot p_2 \cdots p_k</M>
## is a product of <M>-1</M> and of primes, not necessarily distinct,
## and that <M>n</M> is coprime to <M>m</M>.
## Then <M>J(n,m) = J(n,p_1) \cdot J(n,p_2) \cdots J(n,p_k)</M>.
## 
## Note that the Kronecker-Jacobi symbol coincides with the Jacobi symbol
## that is defined for odd <M>m</M> in many number theory books.
## For odd primes <M>m</M> and <M>n</M> coprime to <M>m</M> it coincides
## with the Legendre symbol.
## 
## <Ref Func="Jacobi"/> is very efficient, even for large values of
## <A>n</A> and <A>m</A>, it is about as fast as the Euclidean algorithm
## (see <Ref Func="Gcd" Label="for (a ring and) several elements"/>).
##
## <Example><![CDATA[
## gap> Jacobi( 11, 35 ); # 9^2 = 11 mod 35
## 1
## gap> # this is -1, thus there is no r such that r^2 = 6 mod 35
## gap> Jacobi( 6, 35 );
## -1
## gap> # this is 1 even though there is no r with r^2 = 3 mod 35
## gap> Jacobi( 3, 35 );
## 1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "Jacobi" );

#############################################################################
##
#F Legendre( <n>, <m> ) . . . . . . . . . . . . . . . . . . Legendre symbol
##
## <#GAPDoc Label="Legendre">
## <ManSection>
## <Func Name="Legendre" Arg='n, m'/>
##
## <Description>
## <Index>quadratic residue</Index>
## <Index Subkey="quadratic">residue</Index>
## <Ref Func="Legendre"/> returns the value of the <E>Legendre symbol</E>
## of the integer <A>n</A> modulo the positive integer <A>m</A>.
## 
## The value of the Legendre symbol <M>L(n/m)</M> is 1 if <M>n</M> is a
## <E>quadratic residue</E> modulo <M>m</M>, i.e., if there exists an integer <M>r</M> such
## that <M>r^2 \equiv n \pmod{m}</M> and <M>-1</M> otherwise.
## 
## If a root of <A>n</A> exists it can be found by <Ref Func="RootMod"/>.
## 
## While the value of the Legendre symbol usually is only defined for <A>m</A> a
## prime, we have extended the definition to include composite moduli too.
## The Jacobi symbol (see <Ref Func="Jacobi"/>) is another generalization of the
## Legendre symbol for composite moduli that is much cheaper to compute,
## because it does not need the factorization of <A>m</A> (see <Ref Func="FactorsInt"/>).
## 
## A description of the Jacobi symbol, the Legendre symbol, and related
## topics can be found in <Cite Key="Baker84"/>.
##
## <Example><![CDATA[
## gap> Legendre( 5, 11 ); # 4^2 = 5 mod 11
## 1
## gap> # this is -1, thus there is no r such that r^2 = 6 mod 11
## gap> Legendre( 6, 11 );
## -1
## gap> # this is -1, thus there is no r such that r^2 = 3 mod 35
## gap> Legendre( 3, 35 );
## -1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "Legendre" );

#############################################################################
##
#F RootMod( <n>[, <k>], <m> ) . . . . . . . . . . . root modulo an integer
##
## <#GAPDoc Label="RootMod">
## <ManSection>
## <Func Name="RootMod" Arg='n[, k], m'/>
##
## <Description>
## <Index>quadratic residue</Index>
## <Index Subkey="quadratic">residue</Index>
## <Index Subkey="of an integer modulo another">root</Index>
## <Ref Func="RootMod"/> computes a <A>k</A>th root of the integer <A>n</A>
## modulo the positive integer <A>m</A>,
## i.e., a <M>r</M> such that
## <M>r^{<A>k</A>} \equiv <A>n</A> \pmod{<A>m</A>}</M>.
## If no such root exists <Ref Func="RootMod"/> returns <K>fail</K>.
## If only the arguments <A>n</A> and <A>m</A> are given,
## the default value for <A>k</A> is <M>2</M>.
## 
## A square root of <A>n</A> exists only if <C>Legendre(<A>n</A>,<A>m</A>) = 1</C>
## (see <Ref Func="Legendre"/>).
## If <A>m</A> has <M>r</M> different prime factors then there are <M>2^r</M> different
## roots of <A>n</A> mod <A>m</A>.
## It is unspecified which one <Ref Func="RootMod"/> returns.
## You can, however, use <Ref Func="RootsMod"/> to compute the full set
## of roots.
## 
## <Ref Func="RootMod"/> is efficient even for large values of <A>m</A>,
## in fact the most time is usually spent factoring <A>m</A>
## (see <Ref Func="FactorsInt"/>).
##
## <Example><![CDATA[
## gap> # note 'RootMod' does not return 8 in this case but -8:
## gap> RootMod( 64, 1009 );
## 1001
## gap> RootMod( 64, 3, 1009 );
## 518
## gap> RootMod( 64, 5, 1009 );
## 656
## gap> List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),
## > x -> x mod 1009 ); # set of all square roots of 64 mod 1009
## [ 1001, 8 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RootMod" );

#############################################################################
##
#F RootsMod( <n>[, <k>], <m> ) . . . . . . . . . . . roots modulo an integer
##
## <#GAPDoc Label="RootsMod">
## <ManSection>
## <Func Name="RootsMod" Arg='n[, k], m'/>
##
## <Description>
## <Ref Func="RootsMod"/> computes the set of <A>k</A>th roots of the
## integer <A>n</A> modulo the positive integer <A>m</A>, i.e., the list of
## all <M>r</M> such that <M>r^{<A>k</A>} \equiv <A>n</A> \pmod{<A>m</A>}</M>.
## If only the arguments <A>n</A> and <A>m</A> are given,
## the default value for <A>k</A> is <M>2</M>.
## <Example><![CDATA[
## gap> RootsMod( 1, 7*31 ); # the same as `RootsUnityMod( 7*31 )'
## [ 1, 92, 125, 216 ]
## gap> RootsMod( 7, 7*31 );
## [ 21, 196 ]
## gap> RootsMod( 5, 7*31 );
## [ ]
## gap> RootsMod( 1, 5, 7*31 );
## [ 1, 8, 64, 78, 190 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RootsMod" );

#############################################################################
##
#F RootsUnityMod( [<k>,] <m> ) . . . . . . roots of unity modulo an integer
##
## <#GAPDoc Label="RootsUnityMod">
## <ManSection>
## <Func Name="RootsUnityMod" Arg='[k,] m'/>
##
## <Description>
## <Index>modular roots</Index>
## <Index Subkey="of 1 modulo an integer">root</Index>
## <Ref Func="RootsUnityMod"/> returns the set of <A>k</A>-th roots of unity
## modulo the positive integer <A>m</A>, i.e.,
## the list of all solutions <M>r</M> of
## <M>r^{<A>k</A>} \equiv <A>n</A> \pmod{<A>m</A>}</M>.
## If only the argument <A>m</A> is given,
## the default value for <A>k</A> is <M>2</M>.
## 
## In general there are <M><A>k</A>^n</M> such roots if the modulus <A>m</A>
## has <M>n</M> different prime factors <M>p</M> such that
## <M>p \equiv 1 \pmod{<A>k</A>}</M>.
## If <M><A>k</A>^2</M> divides <A>m</A> then there are
## <M><A>k</A>^{{n+1}}</M> such roots;
## and especially if <M><A>k</A> = 2</M> and 8 divides <A>m</A>
## there are <M>2^{{n+2}}</M> such roots.
## 
## In the current implementation <A>k</A> must be a prime.
## <Example><![CDATA[
## gap> RootsUnityMod( 7*31 ); RootsUnityMod( 3, 7*31 );
## [ 1, 92, 125, 216 ]
## [ 1, 25, 32, 36, 67, 149, 156, 191, 211 ]
## gap> RootsUnityMod( 5, 7*31 );
## [ 1, 8, 64, 78, 190 ]
## gap> List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),
## > x -> x mod 1009 ); # set of all square roots of 64 mod 1009
## [ 1001, 8 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "RootsUnityMod" );

#############################################################################
##
#F LogMod( <n>, <r>, <m> ) . . . . . . discrete logarithm modulo an integer
#F LogModShanks( <n>, <r>, <m> )
##
## <#GAPDoc Label="LogMod">
## <ManSection>
## <Func Name="LogMod" Arg='n, r, m'/>
## <Func Name="LogModShanks" Arg='n, r, m'/>
##
## <Description>
## <Index Subkey="discrete">logarithm</Index>
## computes the discrete <A>r</A>-logarithm of the integer <A>n</A>
## modulo the integer <A>m</A>.
## It returns a number <A>l</A> such that
## <M><A>r</A>^{<A>l</A>} \equiv <A>n</A> \pmod{<A>m</A>}</M>
## if such a number exists.
## Otherwise <K>fail</K> is returned.
## 
## <Ref Func="LogModShanks"/> uses the Baby Step - Giant Step Method
## of Shanks (see for example <Cite Key="Coh93" Where="section 5.4.1"/>)
## and in general requires more memory than a call to <Ref Func="LogMod"/>.
## <Example><![CDATA[
## gap> l:= LogMod( 2, 5, 7 ); 5^l mod 7 = 2;
## 4
## true
## gap> LogMod( 1, 3, 3 ); LogMod( 2, 3, 3 );
## 0
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "LogMod" );
DeclareGlobalFunction( "LogModShanks" );

DeclareGlobalFunction( "DoLogModRho" );

#############################################################################
##
#O Sigma( <n> ) . . . . . . . . . . . . . . . sum of divisors of an integer
##
## <#GAPDoc Label="Sigma">
## <ManSection>
## <Oper Name="Sigma" Arg='n'/>
##
## <Description>
## <Ref Oper="Sigma"/> returns the sum of the positive divisors of the
## nonzero integer <A>n</A>.
## 
## <Ref Oper="Sigma"/> is a multiplicative arithmetic function, i.e.,
## if <M>n</M> and <M>m</M> are relatively prime we have that
## <M>\sigma(n \cdot m) = \sigma(n) \sigma(m)</M>.
## 
## Together with the formula <M>\sigma(p^k) = (p^{{k+1}}-1) / (p-1)</M>
## this allows us to compute <M>\sigma(<A>n</A>)</M>.
## 
## Integers <A>n</A> for which <M>\sigma(<A>n</A>) = 2 <A>n</A></M>
## are called perfect.
## Even perfect integers are exactly of the form
## <M>2^{{<A>n</A>-1}}(2^{<A>n</A>}-1)</M>
## where <M>2^{<A>n</A>}-1</M> is prime.
## Primes of the form <M>2^{<A>n</A>}-1</M> are called
## <E>Mersenne primes</E>, and
## 42 among the known Mersenne primes are obtained for <A>n</A> <M>=</M> 2, 3, 5, 7, 13, 17, 19,
## 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689,
## 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091,
## 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917,
## 20996011, 24036583 and 25964951. Please find more up to date information
## about Mersenne primes at <URL>https://www.mersenne.org</URL>.
## It is not known whether odd perfect integers exist,
## however <Cite Key="BC89"/> show that any such integer must have
## at least 300 decimal digits.
## 
## <Ref Oper="Sigma"/> usually spends most of its time factoring <A>n</A>
## (see <Ref Func="FactorsInt"/>).
## 
## <Example><![CDATA[
## gap> Sigma( 1 );
## 1
## gap> Sigma( 1009 ); # 1009 is a prime
## 1010
## gap> Sigma( 8128 ) = 2*8128; # 8128 is a perfect number
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Sigma", [ IsObject ] );

#############################################################################
##
#O Tau( <n> ) . . . . . . . . . . . . . . number of divisors of an integer
##
## <#GAPDoc Label="Tau">
## <ManSection>
## <Oper Name="Tau" Arg='n'/>
##
## <Description>
## <Ref Oper="Tau"/> returns the number of the positive divisors of the
## nonzero integer <A>n</A>.
## 
## <Ref Oper="Tau"/> is a multiplicative arithmetic function, i.e.,
## if <M>n</M> and <M>m</M> are relative prime we have
## <M>\tau(n \cdot m) = \tau(n) \tau(m)</M>.
## Together with the formula <M>\tau(p^k) = k+1</M> this allows us
## to compute <M>\tau(<A>n</A>)</M>.
## 
## <Ref Oper="Tau"/> usually spends most of its time factoring <A>n</A>
## (see <Ref Func="FactorsInt"/>).
## <Example><![CDATA[
## gap> Tau( 1 );
## 1
## gap> Tau( 1013 ); # thus 1013 is a prime
## 2
## gap> Tau( 8128 );
## 14
## gap> # result is odd if and only if argument is a perfect square:
## gap> Tau( 36 );
## 9
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Tau", [ IsObject ] );

#############################################################################
##
#F MoebiusMu( <n> ) . . . . . . . . . . . . . . Moebius inversion function
##
## <#GAPDoc Label="MoebiusMu">
## <ManSection>
## <Func Name="MoebiusMu" Arg='n'/>
##
## <Description>
## <Ref Func="MoebiusMu"/> computes the value of Moebius inversion function
## for the nonzero integer <A>n</A>.
## This is 0 for integers which are not squarefree, i.e.,
## which are divided by a square <M>r^2</M>.
## Otherwise it is 1 if <A>n</A> has a even number and <M>-1</M> if <A>n</A>
## has an odd number of prime factors.
## 
## The importance of <M>\mu</M> stems from the so called inversion formula.
## Suppose <M>f</M> is a multiplicative arithmetic function
## defined on the positive integers and let
## <M>g(n) = \sum_{{d \mid n}} f(d)</M>.
## Then <M>f(n) = \sum_{{d \mid n}} \mu(d) g(n/d)</M>.
## As a special case we have
## <M>\phi(n) = \sum_{{d \mid n}} \mu(d) n/d</M>
## since <M>n = \sum_{{d \mid n}} \phi(d)</M>
## (see <Ref Oper="Phi"/>).
## 
## <Ref Func="MoebiusMu"/> usually spends all of its time factoring <A>n</A>
## (see <Ref Func="FactorsInt"/>).
## <Example><![CDATA[
## gap> MoebiusMu( 60 ); MoebiusMu( 61 ); MoebiusMu( 62 );
## 0
## -1
## 1
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MoebiusMu" );

#############################################################################
##
#F TwoSquares( <n> ) . . . . . repres. of an integer as a sum of two squares
##
## <#GAPDoc Label="TwoSquares">
## <ManSection>
## <Func Name="TwoSquares" Arg='n'/>
##
## <Description>
## <Index Subkey="as a sum of two squares">representation</Index>
## <Ref Func="TwoSquares"/> returns a list of two integers <M>x \leq y</M>
## such that the sum of the squares of <M>x</M> and <M>y</M> is equal to the
## nonnegative integer <A>n</A>, i.e., <M>n = x^2 + y^2</M>.
## If no such representation exists
## <Ref Func="TwoSquares"/> will return <K>fail</K>.
## <Ref Func="TwoSquares"/> will return a representation for which the gcd
## of <M>x</M> and <M>y</M> is as small as possible.
## It is not specified which representation <Ref Func="TwoSquares"/> returns
## if there is more than one.
## 
## Let <M>a</M> be the product of all maximal powers of primes of the form
## <M>4k+3</M> dividing <A>n</A>.
## A representation of <A>n</A> as a sum of two squares exists
## if and only if <M>a</M> is a perfect square.
## Let <M>b</M> be the maximal power of <M>2</M> dividing <A>n</A> or its
## half, whichever is a perfect square.
## Then the minimal possible gcd of <M>x</M> and <M>y</M> is the square root
## <M>c</M> of <M>a \cdot b</M>.
## The number of different minimal representation with <M>x \leq y</M> is
## <M>2^{{l-1}}</M>, where <M>l</M> is the number of different prime factors
## of the form <M>4k+1</M> of <A>n</A>.
## 
## The algorithm first finds a square root <M>r</M> of <M>-1</M> modulo
## <M><A>n</A> / (a \cdot b)</M>, which must exist,
## and applies the Euclidean algorithm to <M>r</M> and <A>n</A>.
## The first residues in the sequence that are smaller than
## <M>\sqrt{{<A>n</A>/(a \cdot b)}}</M> times <M>c</M> are a possible pair
## <M>x</M> and <M>y</M>.
## 
## Better descriptions of the algorithm and related topics can be found in
## <Cite Key="Wagon90"/> and <Cite Key="Zagier90"/>.
##
## <Example><![CDATA[
## gap> TwoSquares( 5 );
## [ 1, 2 ]
## gap> TwoSquares( 11 ); # there is no representation
## fail
## gap> TwoSquares( 16 );
## [ 0, 4 ]
## gap> # 3 is the minimal possible gcd because 9 divides 45:
## gap> TwoSquares( 45 );
## [ 3, 6 ]
## gap> # it is not [5,10] because their gcd is not minimal:
## gap> TwoSquares( 125 );
## [ 2, 11 ]
## gap> # [10,11] would be the other possible representation:
## gap> TwoSquares( 13*17 );
## [ 5, 14 ]
## gap> TwoSquares( 848654483879497562821 ); # argument is prime
## [ 6305894639, 28440994650 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "TwoSquares" );

#############################################################################
##
#F PValuation( <n>, ) . . . . . . . . . . . prime part exponent
##
## <#GAPDoc Label="PValuation">
## <ManSection>
## <Func Name="PValuation" Arg='n, p'/>
##
## <Description>
## For an integer <A>n</A> and a prime <A>p</A> this function returns
## the <A>p</A>-valuation of <A>n</A>,
## that is the exponent <M>e</M> such that <M>p^e</M> is the largest
## power of <A>p</A> that divides <A>n</A>.
## The valuation of zero is infinity.
##
## <Example><![CDATA[
## gap> PValuation(100,2);
## 2
## gap> PValuation(100,3);
## 0
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PValuation" );

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