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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra gaussfac.spad}
\author{Patrizia Gianni}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package GAUSSFAC GaussianFactorizationPackage}
<<package GAUSSFAC GaussianFactorizationPackage>>=
)abbrev package GAUSSFAC GaussianFactorizationPackage
++ Author: Patrizia Gianni
++ Date Created: Summer 1986
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: Package for the factorization of complex or gaussian
++ integers.
GaussianFactorizationPackage() : C == T
where
NNI == NonNegativeInteger
Z ==> Integer
ZI ==> Complex Z
FRZ ==> Factored ZI
fUnion ==> Union("nil", "sqfr", "irred", "prime")
FFE ==> Record(flg:fUnion, fctr:ZI, xpnt:Integer)
C == with
factor : ZI -> FRZ
++ factor(zi) produces the complete factorization of the complex
++ integer zi.
sumSquares : Z -> List Z
++ sumSquares(p) construct \spad{a} and b such that \spad{a**2+b**2}
++ is equal to
++ the integer prime p, and otherwise returns an error.
++ It will succeed if the prime number p is 2 or congruent to 1
++ mod 4.
prime? : ZI -> Boolean
++ prime?(zi) tests if the complex integer zi is prime.
T == add
import IntegerFactorizationPackage Z
reduction(u:Z,p:Z):Z ==
p=0 => u
positiveRemainder(u,p)
merge(p:Z,q:Z):Union(Z,"failed") ==
p = q => p
p = 0 => q
q = 0 => p
"failed"
exactquo(u:Z,v:Z,p:Z):Union(Z,"failed") ==
p=0 => u exquo v
v rem p = 0 => "failed"
positiveRemainder((extendedEuclidean(v,p,u)::Record(coef1:Z,coef2:Z)).coef1,p)
FMod := ModularRing(Z,Z,reduction,merge,exactquo)
fact2:ZI:= complex(1,1)
---- find the solution of x**2+1 mod q ----
findelt(q:Z) : Z ==
q1:=q-1
r:=q1
r1:=r exquo 4
while not (r1 case "failed") repeat
r:=r1::Z
r1:=r exquo 2
s : FMod := reduce(1,q)
qq1:FMod :=reduce(q1,q)
for i in 2.. while (s=1 or s=qq1) repeat
s:=reduce(i,q)**(r::NNI)
t:=s
while t~=qq1 repeat
s:=t
t:=t**2
s::Z
---- write p, congruent to 1 mod 4, as a sum of two squares ----
sumsq1(p:Z) : List Z ==
s:= findelt(p)
u:=p
while u**2>p repeat
w:=u rem s
u:=s
s:=w
[u,s]
---- factorization of an integer ----
intfactor(n:Z) : Factored ZI ==
lfn:= factor n
r : List FFE :=[]
unity:ZI:=complex(unit lfn,0)
for term in (factorList lfn) repeat
n:=term.fctr
exp:=term.xpnt
n=2 =>
r :=concat(["prime",fact2,2*exp]$FFE,r)
unity:=unity*complex(0,-1)**(exp rem 4)::NNI
(n rem 4) = 3 => r:=concat(["prime",complex(n,0),exp]$FFE,r)
sz:=sumsq1(n)
z:=complex(sz.1,sz.2)
r:=concat(["prime",z,exp]$FFE,
concat(["prime",conjugate(z),exp]$FFE,r))
makeFR(unity,r)
---- factorization of a gaussian number ----
factor(m:ZI) : FRZ ==
m=0 => primeFactor(0,1)
a:= real m
(b:= imag m)=0 => intfactor(a) :: FRZ
a=0 =>
ris:=intfactor(b)
unity:= unit(ris)*complex(0,1)
makeFR(unity,factorList ris)
d:=gcd(a,b)
result : List FFE :=[]
unity:ZI:=1$ZI
if not one? d then
a:=(a exquo d)::Z
b:=(b exquo d)::Z
r:= intfactor(d)
result:=factorList r
unity:=unit r
m:=complex(a,b)
n:Z:=a**2+b**2
factn:= factorList(factor n)
part:FFE:=["prime",0$ZI,0]
for term in factn repeat
n:=term.fctr
exp:=term.xpnt
n=2 =>
part:= ["prime",fact2,exp]$FFE
m:=m quo (fact2**exp:NNI)
result:=concat(part,result)
(n rem 4) = 3 =>
g0:=complex(n,0)
part:= ["prime",g0,exp quo 2]$FFE
m:=m quo g0
result:=concat(part,result)
z:=gcd(m,complex(n,0))
part:= ["prime",z,exp]$FFE
z:=z**(exp:NNI)
m:=m quo z
result:=concat(part,result)
if not one? m then unity:=unity * m
makeFR(unity,result)
---- write p prime like sum of two squares ----
sumSquares(p:Z) : List Z ==
p=2 => [1,1]
not one?(p rem 4) => error "no solutions"
sumsq1(p)
prime?(a:ZI) : Boolean ==
n : Z := norm a
n=0 => false -- zero
n=1 => false -- units
prime?(n)$IntegerPrimesPackage(Z) => true
re : Z := real a
im : Z := imag a
not zero? re and not zero? im => false
p : Z := abs(re+im) -- a is of the form p, -p, %i*p or -%i*p
p rem 4 ~= 3 => false
-- return-value true, if p is a rational prime,
-- and false, otherwise
prime?(p)$IntegerPrimesPackage(Z)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package GAUSSFAC GaussianFactorizationPackage>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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