/usr/share/axiom-20170501/src/algebra/GAUSSFAC.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | )abbrev package GAUSSFAC GaussianFactorizationPackage
++ Author: Patrizia Gianni
++ Date Created: Summer 1986
++ Description:
++ Package for the factorization of complex or gaussian integers.
GaussianFactorizationPackage() : SIG == CODE 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)
SIG ==> 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.
CODE ==> 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 ^(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$ZI
if d^=1 then
a:=(a exquo d)::Z
b:=(b exquo d)::Z
r:= intfactor(d)
result:=factorList r
unity:=unit r
m:=complex(a,b)
n:Z:=a**2+b**2
factn:= factorList(factor n)
part:FFE:=["prime",0$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]$FFE
z:=z**(exp:NNI)
m:=m quo z
result:=concat(part,result)
if m^=1 then unity:=unity * m
makeFR(unity,result)
---- write p prime like sum of two squares ----
sumSquares(p:Z) : List Z ==
p=2 => [1,1]
p rem 4 ^= 1 => error "no solutions"
sumsq1(p)
prime?(a:ZI) : Boolean ==
n : Z := norm a
n=0 => false -- zero
n=1 => false -- units
prime?(n)$IntegerPrimesPackage(Z) => true
re : Z := real a
im : Z := imag a
re^=0 and im^=0 => 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)
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