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MathGroup Archive 1995

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Factorization package

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg803] Factorization package
  • From: Dave Wagner <wagner at bullwinkle.cs.Colorado.EDU>
  • Date: Fri, 21 Apr 1995 01:42:24 -0400

Okay group, here's the package that I promised.  Note that this is a
"Call for Comments."  Please don't clutter up the net/mailing list with
comments directed to me; e-mail them to me directly (dbwagner at csn.net)
and I will summarize all at once.

Some background:

Alan Powell (powella at delphi.com) asked:
>>Could anyone suggest a neat way to reformat the output of
>>FactorInteger into a more usable print format?
>< details deleted >

To which I replied that the best thing to do would be to convert
an expression like {{2,2}},{3,4}} (the result of FactorInteger)
to something like HoldForm[Times[Power[2,2], Power[3,4]]],
which would print as

 2  4
2  3

Later I posted another message saying that this was doing things the
hard way, and that all that was necessary was to define a new head
called IntegerPower like this:

	Format[IntegerPower[a_,b_]] := HoldForm[Power[a,b]]

and then convert the factorization shown above to
Times[IntegerPower[2,2], IntegerPower[3,4]].
I noted that it wasn't really necessary to have a special head for
expressions like this.

WELL, I WAS WRONG, inasmuch as if you want to try to make these
beasties behave like a built-in type, it is much easier if you give
them a readily-identifiable head.  In particular, without such a
head there are two possible representations of a factorization:
Time[__IntegerPower] and IntegerPower[_,_] (for numbers with only
one prime factor).  This complicates things enormously.

So, I've written a package that defines a new "type" called
Factorization, which looks simply like Factorization[{2,2}, {3,4}].
You can add, multiply, and exponentiate factorizations together
with other numbers to your hearts' desire.  In addition, the
built-in function FactorInteger is overridden to return a
Factorization directly.  I've also defined an inverse function,
ExpandInteger.  Here are some examples:

f1 = FactorInteger[3^2]
 2
3

f1 + 4^2
 2
5

% + 1325
   3  2
2 3  5

% ^ 2
 2  6  4
2  3  5

1/%
 -2  -6  -4
2   3   5

f2 = FactorInteger[238500]
 2  2  3
2  3  5  53

Sqrt[f2]
     3/2   1/2
2 3 5    53

N[%]
488.365

f4 = FactorInteger[40 + 18I]
          2
-1 (1 + I)  (1 + 6 I) (3 + 2 I)

f4 + 1
-I (1 + 20 I) (2 + I)

ExpandInteger[%]
41 + 18 I

f4 + .5
40.5 + 18 I

Complex numbers are kind of a thorny problem -- when I add,
for example, 2^2 + 4+I I don't know whether it would be better
to return 2^3 + I or FactorInteger[8+I] = -I (1+2I) (3+2I).
Currently I do the latter; I've thought of perhaps doing
the former and overriding, say, Together, to go the rest of
the way.  But I hesitate to pollute the definition of Together
like that.  Comments?

I've made a start on UpValues for all of the numerical functions, but
there are so many that I decided to take a rest for a couple of days.
Here is an example of one of my better efforts:

FactorInteger[7/9]
 -2
3   7

Cos[% Pi]
        -2
Cos[Pi 3   7]

FactorInteger[7/6]
 -1  -1
2   3   7

Cos[% Pi]
-Sqrt[3]
--------
   2

Be advised that some of the built-in functions that I haven't gotten to
yet may hit $RecursionLimit as a result of the overridden definitions of
Plus and Times.  ComplexExpand, Arg, and Abs were like that until I
fixed them.  Unfortunately, I'm working alphabetically and I'm only up
to "Delete"!

In the meantime, I was hoping that members of the group would
beat on this package and try to find bugs (keeping in mind the
disclaimer in the previous paragraph).  I particularly would
like input on design issues, such as when I should and should not
assume the user wants a factorization converted to a number or
vice-versa.

The package is appended below in notebook format.  To use it, you can
open it directly and reply "Yes" to "Evaluate Initialization Cells?".
Or, save it in a file called "Factorization.m", put the file in a place
where Mathematica will find it (see $Path), and evaluate
Needs["Factorization`"].  There are some examples of use at the
beginning of the notebook.

		Dave Wagner
		Principia Consulting
		(303) 786-8371
		princon at csn.net
		http://www.csn.net/princon

----------------- CLIP HERE and save in "Factorization.m" -------------------
(*^
::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
	transferred by email, ftp, or other text-file transfer utility.  It should
	be read or edited using a copy of Mathematica or MathReader.  If you 
	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

	FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2";

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	fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, e8,  24, "Times"; 
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	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M27,  12, "Times"; 
	paletteColors = 128; showRuler; automaticGrouping; currentKernel; 
]
:[font = title; inactive; preserveAspect; startGroup]
Factorization` Abstract Data Type Package
:[font = section; inactive; preserveAspect; startGroup]
Demonstration of use
:[font = input; preserveAspect; startGroup]
?FactorInteger
:[font = print; inactive; preserveAspect; endGroup]
FactorInteger[n] gives a list of the prime factors of
   the integer n, together with their exponents.  The
   returned list is wrapped in the     head
   Factorization.
:[font = input; preserveAspect; startGroup]
?Factorization
:[font = print; inactive; preserveAspect; endGroup]
Factorization[{n1, p1}, {n2, p2}, ...] represents  the
   number n1^p1 * n2^p2 * ...
:[font = input; preserveAspect; startGroup]
f1 = FactorInteger[3^2]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{3, 2}]
;[o]
 2
3
:[font = input; preserveAspect; startGroup]
f1 + 4^2
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{5, 2}]
;[o]
 2
5
:[font = input; preserveAspect; startGroup]
% + 1325
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{2, 1}, {3, 3}, {5, 2}]
;[o]
   3  2
2 3  5
:[font = input; preserveAspect; startGroup]
% ^ 2
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{2, 2}, {3, 6}, {5, 4}]
;[o]
 2  6  4
2  3  5
:[font = input; preserveAspect; startGroup]
1/%
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{2, -2}, {3, -6}, {5, -4}]
;[o]
 -2  -6  -4
2   3   5
:[font = input; preserveAspect; startGroup]
Sqrt[%]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{2, -1}, {3, -3}, {5, -2}]
;[o]
 -1  -3  -2
2   3   5
:[font = input; preserveAspect; startGroup]
ExpandInteger[%]
:[font = output; output; inactive; preserveAspect; endGroup]
1/1350
;[o]
 1
----
1350
:[font = text; inactive; preserveAspect]
Here is an example in which a factorization, through manipulation, no longer represents a rational number.
:[font = input; preserveAspect; startGroup]
f2 = FactorInteger[238500]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{2, 2}, {3, 2}, {5, 3}, {53, 1}]
;[o]
 2  2  3
2  3  5  53
:[font = input; preserveAspect; startGroup]
Sqrt[f2]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{2, 1}, {3, 1}, {5, 3/2}, {53, 1/2}]
;[o]
     3/2   1/2
2 3 5    53
:[font = text; inactive; preserveAspect]
Note that a Factorization can be numericalized without first converting it back to a number.
:[font = input; preserveAspect; startGroup]
N[%]
:[font = output; output; inactive; preserveAspect; endGroup]
488.3646178829912
;[o]
488.365
:[font = input; preserveAspect; startGroup]
ExpandInteger[Sqrt[f2]]
:[font = output; output; inactive; preserveAspect; endGroup]
30*265^(1/2)
;[o]
30 Sqrt[265]
:[font = input; preserveAspect; startGroup]
N[%]
:[font = output; output; inactive; preserveAspect; endGroup]
488.3646178829912
;[o]
488.365
:[font = text; inactive; preserveAspect]
Factors of -1 are always stand alone in a factorization, and are combined as necessary.
:[font = input; preserveAspect; startGroup]
f3 = FactorInteger[-3]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{-1, 1}, {3, 1}]
;[o]
-1 3
:[font = input; preserveAspect; startGroup]
f3 ^ 2
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{3, 2}]
;[o]
 2
3
:[font = input; preserveAspect; startGroup]
Sqrt[f3]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{-1, 1/2}, {3, 1/2}]
;[o]
  1/2  1/2
-1    3
:[font = input; preserveAspect; startGroup]
ExpandInteger[%]
:[font = output; output; inactive; preserveAspect; endGroup]
I*3^(1/2)
;[o]
I Sqrt[3]
:[font = text; inactive; preserveAspect]
Complex numbers are factored over the Gaussian integers.
:[font = input; preserveAspect; startGroup]
f4 = FactorInteger[40 + 18I]
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{-1, 1}, {1 + I, 2}, {1 + 6*I, 1}, 
 
  {3 + 2*I, 1}]
;[o]
          2
-1 (1 + I)  (1 + 6 I) (3 + 2 I)
:[font = input; preserveAspect; startGroup]
f4 + 1
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{-I, 1}, {1 + 20*I, 1}, {2 + I, 1}]
;[o]
-I (1 + 20 I) (2 + I)
:[font = input; preserveAspect; startGroup]
ExpandInteger[%]
:[font = output; output; inactive; preserveAspect; endGroup]
41 + 18*I
;[o]
41 + 18 I
:[font = text; inactive; preserveAspect]
Adding an inexact number to any factorization numericalizes it.
:[font = input; preserveAspect; startGroup]
f4 + .5
:[font = output; output; inactive; preserveAspect; endGroup]
40.5 + 18*I
;[o]
40.5 + 18 I
:[font = input; preserveAspect; startGroup]
f1 + .5
:[font = output; output; inactive; preserveAspect; endGroup]
9.5
;[o]
9.5
:[font = text; inactive; preserveAspect]
Powers can be made in any combination, except that when a factorization is used as aexponent, the exponent is first expanded.
:[font = input; preserveAspect; startGroup]
{f1, f3}
:[font = output; output; inactive; preserveAspect; endGroup]
{Factorization[{3, 2}], 
 
  Factorization[{-1, 1}, {3, 1}]}
;[o]
  2
{3 , -1 3}
:[font = input; preserveAspect; startGroup]
f1 ^ f3
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{3, -6}]
;[o]
 -6
3
:[font = input; preserveAspect; startGroup]
f3 ^ f1
:[font = output; output; inactive; preserveAspect; endGroup]
Factorization[{-1, 1}, {3, 9}]
;[o]
    9
-1 3
:[font = input; preserveAspect; startGroup]
f1 ^ 2.5
:[font = output; output; inactive; preserveAspect; endGroup]
243.
;[o]
243.
:[font = input; preserveAspect; startGroup]
2.5 ^ f1
:[font = output; output; inactive; preserveAspect; endGroup]
3814.697265625
;[o]
3814.7
:[font = input; preserveAspect; startGroup]
2 ^ f1
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Factorization[{2, 9}]
;[o]
 9
2
:[font = section; inactive; initialization; preserveAspect; startGroup]
Implementation
:[font = subsection; inactive; preserveAspect; startGroup]
Public part of package
:[font = input; initialization; preserveAspect]
*)
BeginPackage["Factorization`"];
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Usage messages
:[font = input; initialization; preserveAspect]
*)
Factorization::usage =
"Factorization[{n1, p1}, {n2, p2}, ...] represents
 the number n1^p1 * n2^p2 * ...";
(*
:[font = input; initialization; preserveAspect]
*)
If[StringPosition[FactorInteger::usage,
	"Factorization"] === {},
 
	FactorInteger::usage = FactorInteger::usage <>
	" " <> " The returned list is wrapped in the
    head Factorization."
];
(*
:[font = input; initialization; preserveAspect]
*)
ExpandInteger::usage =
"ExpandInteger[factzn] converts the Factorization
 factzn to an integer, if possible.";
(*
:[font = input; initialization; preserveAspect; endGroup; endGroup]
*)
ExpandAllInteger::usage =
"ExpandAllInteger[expr] maps ExpandInteger onto
 every part of expr.";
(*
:[font = subsection; inactive; preserveAspect; startGroup]
Private part of package
:[font = input; initialization; preserveAspect]
*)
Begin["`Private`"];
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
FormatValues
:[font = input; initialization; preserveAspect; startGroup]
*)
fmtrule1 = {
	c:Complex[x_ /; x!=0, y_ /; y!=0] ->
		SequenceForm["(", c, ")"]
}
(*
:[font = output; output; inactive; initialization; preserveAspect; endGroup]
{c:Complex[x_ /; x != 0, y_ /; y != 0] -> "("c")"}
;[o]
{c:Complex[x_ /; x != 0, y_ /; y != 0] -> (c)}
:[font = input; initialization; preserveAspect]
*)
fmtrule2 = {
	{a_, 1} -> SequenceForm[a, " "],
	{a_, b_} -> SequenceForm[a, Superscript[b], " "]
};
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
Format[x_Factorization] :=
	(SequenceForm @@ x) /. fmtrule1 /. fmtrule2 //
		Flatten // Drop[#, -1]&
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Simplification of special cases
:[font = input; initialization; preserveAspect]
*)
Factorization[] := 1
(*
:[font = input; initialization; preserveAspect]
*)
Factorization[{a_, 1}] := a
(*
:[font = input; initialization; preserveAspect]
*)
Factorization[h___, {x_, 0}, t___] :=
	Factorization[h, t]
(*
:[font = input; initialization; preserveAspect]
*)
Factorization[h___, {-1, n_?OddQ /; n>1}, t___] :=
	Factorization[h, {-1, 1}, t]
(*
:[font = input; initialization; preserveAspect]
*)
Factorization[h___, {-1, n_?EvenQ}, t___] :=
	Factorization[h, t]
(*
:[font = input; initialization; preserveAspect]
*)
SetAttributes[Factorization, Orderless]
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
Factorization[h___, {a_, b_}, {a_, c_}, t___] :=
	Factorization[h, {a, b+c}, t]
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Expanding factorizations
:[font = input; initialization; preserveAspect]
*)
ExpandInteger[x_Factorization] :=
	Times @@ Apply[Power, x, {1}]
(*
:[font = input; initialization; preserveAspect]
*)
ExpandInteger[x_] := x
(*
:[font = text; inactive; preserveAspect]
This rule is used by the addition rules that come later.
:[font = input; initialization; preserveAspect]
*)
ExpandInteger[x__] :=
	Sequence @@ ExpandInteger /@ {x}
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
ExpandAllInteger[x_] :=
	MapAll[ExpandInteger, x]
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
NValues
:[font = input; initialization; preserveAspect]
*)
N[x_Factorization, ___] := ExpandInteger[x]
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
wasProtected = Unprotect[NumberQ];
NumberQ[_Factorization] := True
Protect @@ wasProtected;
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Utility functions
:[font = input; initialization; preserveAspect]
*)
exact[x_] :=
	NumberQ[x] && (Precision[x] == Infinity)
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
inexact[x_] :=
	NumberQ[x] && (Precision[x] < Infinity)
(*
:[font = subsubsection; inactive; initialization; preserveAspect; startGroup]
Overriding FactorInteger
:[font = input; initialization; preserveAspect]
*)
wasProtected = Unprotect[FactorInteger];
(*
:[font = input; initialization; preserveAspect]
*)
FactorInteger[x_Factorization] := x
(*
:[font = input; initialization; preserveAspect]
*)
intercept = True;
FactorInteger[x_?exact, opts___Rule] /; intercept :=
	Block[{intercept = False},	(* no recursion *)
		Factorization @@ FactorInteger[x, opts]
	]
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
Protect @@ wasProtected;
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Multiplication rules
:[font = input; initialization; preserveAspect]
*)
Factorization /:
x_Factorization * y_Factorization := Join[x, y]
(*
:[font = input; initialization; preserveAspect]
*)
Factorization /:
Times[x_Factorization, y_?inexact] := N[x] y
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
Factorization /:
Times[x_Factorization, y_?exact /; y!=0] :=
	Factorization @@
		Block[{intercept = False},
			Join[List @@ x, FactorInteger[y]]
		]
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Rules for exponentiation
:[font = input; initialization; preserveAspect]
*)
Factorization /:
Power[x_?exact, y_Factorization] :=
	FactorInteger[x^ExpandInteger[y]]
(*
:[font = input; initialization; preserveAspect]
*)
Factorization /:
Power[x_, y_Factorization] := x^ExpandInteger[y]
(*
:[font = input; initialization; preserveAspect]
*)
Factorization /:
Power[x_Factorization, y_?inexact] := N[x]^y
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
Factorization /:
Power[x_Factorization, y_?exact] :=
	x /. {n_, p_} -> {n, p y}
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Addition rules
:[font = input; initialization; preserveAspect]
*)
Factorization /:
Plus[x_?inexact, y__Factorization] :=
	Plus[x, ExpandInteger[y]]
(*
:[font = input; initialization; preserveAspect; endGroup; endGroup]
*)
Factorization /:
Plus[x_?exact, y__Factorization] :=
	FactorInteger[Plus[ExpandInteger[x, y]]]
(*
:[font = subsection; inactive; preserveAspect; startGroup]
The following two sections are "under construction"
:[font = subsubsection; inactive; preserveAspect; startGroup]
Miscellaneous numerical functions
:[font = text; inactive; preserveAspect]
UpValues for Factorization
:[font = input; initialization; preserveAspect]
*)
Abs[x_Factorization] ^:= Abs[ExpandInteger[x]]
(*
:[font = input; initialization; preserveAspect]
*)
Arg[x_Factorization] ^:= Arg[ExpandInteger[x]]
(*
:[font = input; initialization; preserveAspect]
*)
BernoulliB[x_Factorization] ^:=
	BernoulliB[ExpandInteger[x]]
BernoulliB[n_Factorization, x_] ^:=
	BernoulliB[ExpandInteger[n], x]
(*
:[font = input; initialization; preserveAspect]
*)
BesselI[n_Factorization, z_?inexact] ^:=
	BesselI[ExpandInteger[n], z]
BesselJ[n_Factorization, z_?inexact] ^:=
	BesselJ[ExpandInteger[n], z]
BesselK[n_Factorization, z_?inexact] ^:=
	BesselK[ExpandInteger[n], z]
BesselY[n_Factorization, z_?inexact] ^:=
	BesselY[ExpandInteger[n], z]
(*
:[font = input; initialization; preserveAspect]
*)
Beta[a_Factorization, b_] ^:=
	Beta[ExpandInteger[a], b]
Beta[a_, b_Factorization] ^:=
	Beta[a, ExpandInteger[b]]
(*
:[font = input; initialization; preserveAspect]
*)
BetaRegularized[z_?inexact, a_Factorization, b_?NumberQ] ^:=
	BetaRegularized[z, ExpandInteger[a], b]
BetaRegularized[z_?inexact, a_?NumberQ, b_Factorization] ^:=
	BetaRegularized[z, a, ExpandInteger[b]]
(*
:[font = input; initialization; preserveAspect]
*)
Binomial[n_Factorization, m_?NumberQ] ^:=
	Binomial[ExpandInteger[n], m]
Binomial[n_, m_Factorization] ^:=
	Binomial[n, ExpandInteger[m]]
(*
:[font = input; initialization; preserveAspect]
*)
Ceiling[x_Factorization] ^:=
	Ceiling[ExpandInteger[x]]
(*
:[font = input; initialization; preserveAspect]
*)
ChebyshevT[n_Factorization, x_] ^:=
	ChebyshevT[ExpandInteger[n], x]
ChebyshevU[n_Factorization, x_] ^:=
	ChebyshevU[ExpandInteger[n], x]
(*
:[font = input; initialization; preserveAspect]
*)
Conjugate[z_Factorization] ^:=
	FactorInteger[Conjugate[ExpandInteger[z]]]
(*
:[font = input; initialization; preserveAspect]
*)
Cyclotomic[n_Factorization, x_] ^:=
	Cyclotomic[ExpandInteger[n], x]
(*
:[font = input; initialization; preserveAspect]
*)
Factorization /:
Derivative[h___, n_Factorization, t___] :=
	Derivative[h, ExpandInteger[n], t]
(*
:[font = input; initialization; preserveAspect]
*)
Divisors[x_Factorization] ^:=
	Divisors[ExpandInteger[x]]
(*
:[font = input; initialization; preserveAspect; endGroup]
*)
Factorization /:
DivisorSigma[k_Factorization, x_Integer] :=
	FactorInteger[DivisorSigma[ExpandInteger[k], x]]
Factorization /:
DivisorSigma[k_Integer, x_Factorization] :=
	FactorInteger[DivisorSigma[k, ExpandInteger[x]]]
Factorization /:
DivisorSigma[k_Factorization, x_Factorization] :=
	FactorInteger[DivisorSigma[ExpandInteger[k, x]]]
(*
:[font = subsubsection; inactive; preserveAspect; startGroup]
Other functions
:[font = input; initialization; preserveAspect]
*)
wasProtected = Unprotect[ComplexExpand, D, Delete,
	Cos, Sin, Sec, Csc, Tan, Cot];
(*
:[font = input; initialization; preserveAspect]
*)
D[f_, h___, {x_, n_Factorization}, t___] :=
	D[f, h, {x, ExpandInteger[n]}, t]
(*
:[font = input; initialization; preserveAspect]
*)
Delete[expr_, n_ /; !FreeQ[n, Factorization]] :=
	Delete[expr, ExpandAllInteger[n]]
(*
:[font = input; initialization; preserveAspect]
*)
ComplexExpand[x_ /; !FreeQ[x, Factorization]] :=
	ComplexExpand[ExpandAllInteger[x]]
(*
:[font = input; initialization; preserveAspect]
*)
Cos[x_Factorization Pi] := Cos[ExpandInteger[x] Pi] /;
	With[{y = ExpandInteger[x]},
		 IntegerQ[2y] || IntegerQ[3y] ||
		 IntegerQ[4y] || IntegerQ[6y]
	]
(*
:[font = input; initialization; preserveAspect]
*)
Sin[x_Factorization Pi] := Sin[ExpandInteger[x] Pi] /;
	With[{y = ExpandInteger[x]},
		 IntegerQ[2y] || IntegerQ[3y] ||
		 IntegerQ[4y] || IntegerQ[6y]
	]
(*
:[font = input; initialization; preserveAspect]
*)
Sec[x_Factorization Pi] := Sec[ExpandInteger[x] Pi] /;
	With[{y = ExpandInteger[x]},
		 IntegerQ[2y] || IntegerQ[3y] ||
		 IntegerQ[4y] || IntegerQ[6y]
	]
(*
:[font = input; initialization; preserveAspect]
*)
Csc[x_Factorization Pi] := Csc[ExpandInteger[x] Pi] /;
	With[{y = ExpandInteger[x]},
		 IntegerQ[2y] || IntegerQ[3y] ||
		 IntegerQ[4y] || IntegerQ[6y]
	]
(*
:[font = input; initialization; preserveAspect]
*)
Tan[x_Factorization Pi] := Tan[ExpandInteger[x] Pi] /;
	With[{y = ExpandInteger[x]},
		 IntegerQ[2y] || IntegerQ[3y] ||
		 IntegerQ[4y] || IntegerQ[6y]
	]
(*
:[font = input; initialization; preserveAspect]
*)
Cot[x_Factorization Pi] := Cot[ExpandInteger[x] Pi] /;
	With[{y = ExpandInteger[x]},
		 IntegerQ[2y] || IntegerQ[3y] ||
		 IntegerQ[4y] || IntegerQ[6y]
	]
(*
:[font = input; initialization; preserveAspect; endGroup; endGroup]
*)
Protect @@ wasProtected;
(*
:[font = subsection; inactive; preserveAspect; startGroup]
Cleanup
:[font = input; initialization; preserveAspect]
*)
End[];
(*
:[font = input; initialization; preserveAspect]
*)
Protect[Factorization];
(*
:[font = input; initialization; preserveAspect; endGroup; endGroup; endGroup]
*)
EndPackage[]
(*
^*)


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