9.22 Factored

Factored creates a domain whose objects are kept in factored form as long as possible. Thus certain operations like * (multiplication) and gcdgcdFactored are relatively easy to do. Others, such as addition, require somewhat more work, and the result may not be completely factored unless the argument domain R provides a factorfactorFactored operation. Each object consists of a unit and a list of factors, where each factor consists of a member of R (the base), an exponent, and a flag indicating what is known about the base. A flag may be one of nil, sqfr, irred or prime, which mean that nothing is known about the base, it is square-free, it is irreducible, or it is prime, respectively. The current restriction to factored objects of integral domains allows simplification to be performed without worrying about multiplication order.

9.22.1 Decomposing Factored Objects

In this section we will work with a factored integer.

g := factor(4312)

\[\]

237211

_{Type: Factored Integer}

Let’s begin by decomposing g into pieces. The only possible units for integers are 1 and -1.

unit(g)

\[\]

1

_{Type: PositiveInteger}

There are three factors.

numberOfFactors(g)

\[\]

3

_{Type: PositiveInteger}

We can make a list of the bases, ...

[nthFactor(g,i) for i in 1..numberOfFactors(g)]

\[\]

[2,7,11]

_{Type: List Integer}

and the exponents, ...

[nthExponent(g,i) for i in 1..numberOfFactors(g)]

\[\]

[3,2,1]

_{Type: List Integer}

and the flags. You can see that all the bases (factors) are prime.

[nthFlag(g,i) for i in 1..numberOfFactors(g)]

\[\]

[“prime”,”prime”,”prime”]

_{Type: List Union(“nil”,”sqfr”,”irred”,”prime”)}

A useful operation for pulling apart a factored object into a list of records of the components is factorListfactorListFactored.

factorList(g)

\[\]

[[flg=”prime”,fctr=2,xpnt=3],[flg=”prime”,fctr=7,xpnt=2],[flg=”prime”,fctr=11,xpnt=1]]

_{Type: List Record(flg: Union(“nil”,”sqfr”,”irred”,”prime”), fctr:} Integer,xpnt: Integer)

If you don’t care about the flags, use factorsfactorsFactored.

factors(g)

\[\]

[[factor=2,exponent=3],[factor=7,exponent=2],[factor=11,exponent=1]]

_{Type: List Record(factor: Integer,exponent: Integer)}

Neither of these operations returns the unit.

first(%).factor

\[\]

2

_{Type: PositiveInteger}

9.22.2 Expanding Factored Objects

Recall that we are working with this factored integer.

g := factor(4312)

\[\]

237211

_{Type: Factored Integer}

To multiply out the factors with their multiplicities, use expandexpandFactored.

expand(g)

\[\]

4312

_{Type: PositiveInteger}

If you would like, say, the distinct factors multiplied together but with multiplicity one, you could do it this way.

reduce(*,[t.factor for t in factors(g)])

\[\]

154

_{Type: PositiveInteger}

9.22.3 Arithmetic with Factored Objects

We’re still working with this factored integer.

g := factor(4312)

\[\]

237211

_{Type: Factored Integer}

We’ll also define this factored integer.

f := factor(246960)

\[\]

2432573

_{Type: Factored Integer}

Operations involving multiplication and division are particularly easy with factored objects.

f * g

\[\]

273257511

_{Type: Factored Integer}

f^500

\[\]

2200031000550071500

_{Type: Factored Integer}

gcd(f,g)

\[\]

2372

_{Type: Factored Integer}

lcm(f,g)

\[\]

243257311

_{Type: Factored Integer}

If we use addition and subtraction things can slow down because we may need to compute greatest common divisors.

f + g

\[\]

2372641

_{Type: Factored Integer}

f - g

\[\]

2372619

_{Type: Factored Integer}

Test for equality with 0 and 1 by using zero?zero?Factored and one?one?Factored, respectively.

zero?(factor(0))

\[\]

true

_{Type: Boolean}

zero?(g)

\[\]

false

_{Type: Boolean}

one?(factor(1))

\[\]

true

_{Type: Boolean}

one?(f)

\[\]

false

_{Type: Boolean}

Another way to get the zero and one factored objects is to use package calling (see ugTypesPkgCallPage in Section ugTypesPkgCallNumber ).

0$Factored(Integer)

\[\]

0

_{Type: Factored Integer}

1$Factored(Integer)

\[\]

1

_{Type: Factored Integer}

9.22.4 Creating New Factored Objects

The mapmapFactored operation is used to iterate across the unit and bases of a factored object. See FactoredFunctionsTwoXmpPage for a discussion of mapmapFactored.

The following four operations take a base and an exponent and create a factored object. They differ in handling the flag component.

nilFactor(24,2)

\[\]

242

_{Type: Factored Integer}

This factor has no associated information.

nthFlag(%,1)

\[\]

“nil”

_{Type: Union(“nil”,...)}

This factor is asserted to be square-free.

sqfrFactor(30,2)

\[\]

302

_{Type: Factored Integer}

This factor is asserted to be irreducible.

irreducibleFactor(13,10)

\[\]

1310

_{Type: Factored Integer}

This factor is asserted to be prime.

primeFactor(11,5)

\[\]

115

_{Type: Factored Integer}

A partial inverse to factorListfactorListFactored is makeFRmakeFRFactored.

h := factor(-720)

\[\]

-24325

_{Type: Factored Integer}

The first argument is the unit and the second is a list of records as returned by factorListfactorListFactored.

h - makeFR(unit(h),factorList(h))

\[\]

0

_{Type: Factored Integer}

9.22.5 Factored Objects with Variables

Some of the operations available for polynomials are also available for factored polynomials.

p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x +

63

\[\]

(4x2-12x+9)y2+(4x2-12x+9)y+28x2-84x+63

_{Type: Polynomial Integer}

fp := factor(p)

\[\]

(2x-3)2(y2+y+7)

_{Type: Factored Polynomial Integer}

You can differentiate with respect to a variable.

D(p,x)

\[\]

(8x-12)y2+(8x-12)y+56x-84

_{Type: Polynomial Integer}

D(fp,x)

\[\]

4(2x-3)(y2+y+7)

_{Type: Factored Polynomial Integer}

numberOfFactors(%)

\[\]

3

_{Type: PositiveInteger}