Re: FactorInteger Print Formatting

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg758] Re: FactorInteger Print Formatting
• From: wagner at bullwinkle.cs.Colorado.EDU (Dave Wagner)
• Date: Tue, 18 Apr 1995 00:48:59 -0400
• Organization: University of Colorado, Boulder

```>   Alan Powell <POWELLA at delphi.com> wrote:
>->
>->
>->Could anyone suggest a neat way to reformat the output of
>->FactorInteger into a more usable print format?
>->

Hey group,

We're all spinning our wheels on this, because we (and that includes
me) all made the assumption that the right way to proceed is to define
some kind of special print form for the entire list returned by
FactorInteger.  That's the hard way to do it!  All that's necessary is
to define a head that prints a single pair of numbers in the form
base^exponent; Mathematica does the rest.  Then you can turn the result
of FactorInteger into something that's truly useful:

In[1]:=
factors = FactorInteger[238500]
Out[1]=
{{2, 2}, {3, 2}, {5, 3}, {53, 1}}

In[2]:=
Format[IntegerPower[x_, y_]] := HoldForm[x^y]

In[4]:=
Apply[IntegerPower, factors, {1}]
Out[4]=
2   2   3    1
{2 , 3 , 5 , 53 }

Now if we apply Times to this list we'll get something that appears to be a
product of powers:

In[5]:=
factorization = Times @@ %
Out[5]=
2  2  3   1
2  3  5  53

But in reality it's a product of IntegerPowers, which is why it doesn't
evaluate to an integer.

In[6]:=
FullForm[%]
Out[6]//FullForm=
Times[IntegerPower[2, 2], IntegerPower[3, 2],
IntegerPower[5, 3], IntegerPower[53, 1]]

So far, so good! Now we need to create a function that goes the other way:

In[21]:=
ExpandInteger[x:(a_IntegerPower * b_.)] :=
x /. IntegerPower->Power
ExpandInteger[x_] := x

In[23]:=
ExpandInteger[factorization]
Out[23]=
238500

In[24]:=
ExpandInteger[%]
Out[24]=
238500

This is just the tip of the iceberg.  I'm working on a package that, I hope,
will let one compute using these factorizations in much the same way
as one uses integers.  It's not in a finished form yet, but when it is
I'll post it.  Meanwhile, to whet your appetite, here are some examples
of functionality that I've already implemented:

In[44]:=
x = FactorInteger[384]
Out[44]=
7  1
2  3

In[75]:=
N[x]
Out[75]=
384.

In[45]:=
y = FactorInteger[36]
Out[45]=
2  2
2  3

In[46]:=
x y
Out[46]=
9  3
2  3

In[55]:=
y / x
Out[55]=
-5  1
2   3

In[61]:=
FactorInteger[238500] / FactorInteger[40950]
Out[61]=
1  1  -1   -1   1
2  5  7   13   53

In[62]:=
1/%
Out[62]=
-1  -1  1   1   -1
2   5   7  13  53

In[63]:=
ExpandInteger[%]
Out[63]=
91
---
530

In[64]:=
40950/238500
Out[64]=
91
---
530

In[65]:=
FactorInteger[96]
Out[65]=
5  1
2  3

In[66]:=
%^3
Out[66]=
15  3
2   3

In[67]:=
% ^ (1/4)
Out[67]=
15  3 1/4
(2   3 )

In[68]:=
PowerExpand[%]
Out[68]=
15/4  3/4
2     3

In[69]:=
ExpandInteger[%]
Out[69]=
3/4
8 6

By now you probably all think I have *way* too much time on my hands,
but the truth is that I'm writing a book and this was just the example
I was looking for!  This group is terrific for providing examples that
are interesting and educational.

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

```

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