Factor bothering - Part 1

So let’s start, what's a factor? Basically it's a number (other than 1) that you multiply with another factor to get an integer.

E.g. 2*2=4 so 2 is a factor of 4, 2*6=12, 3*4=12 so 2,6,3 and 4 are factors of 12.

What I would like to do is have a look for any patterns in these factors, if any.

We need to generate some factors first. I’ve created a little Perl script to do that:

factorGenerator.pl my $intToBeFactored=0; my $intTrialDivisor=0; my $intMaxTrialDivisor=0; for($intToBeFactored=1 ; $intToBeFactored<=$ARGV[0] ; $intToBeFactored++) { $intMaxTrialDivisor=0.5+sqrt($intToBeFactored); for($intTrialDivisor=2 ; $intTrialDivisor<$intMaxTrialDivisor ; $intTrialDivisor++) { if($intToBeFactored % $intTrialDivisor==0) { print("$intTrialDivisor ".$intToBeFactored/$intTrialDivisor." "); } } print "\n"; }

The script is based on the trial divisor algorithm mainly because all the other ones did my head in.

Here is some example output:

C:\stuff\FactorBothering\Part1>factorGenerator.pl 12 2 2 2 3 2 4 3 3 2 5 2 6 3 4

(Yes, I'm on a windows machine)

Each row shows the factors for that row count, so row 4 shows "2 2" while row 5, a prime, shows nothing.

I'm committing heresy here by not including 1, but I'll live with it.

Now we'll do some pretty stuff.

First we need a good number of factors, say 1000.

factorGenerator.pl 1000 > factors1000.txt

Have a look in factors1000.txt, we're going to plot that graphically now.

I've coded a Python script (Perl's graphics libraries aren't worth the effort) that reads the above file and plots a pixel for each factor and marks out the prime numbers:

plotFactors.py import Image, ImageDraw import os size = 1000,1001 im=Image.new("1",size) draw=ImageDraw.Draw(im) draw.rectangle(((0,0),size),1) #set image bg to white lineCount=0 factorFile=open('factors1000.txt','r') for line in factorFile: lineCount=lineCount+1 isPrime=1 for factor in line.split(): im.putpixel((int(factor)+10,lineCount),0) isPrime=0 if isPrime: draw.line(((4,lineCount),(6,lineCount)),0) factorFile.close() im.save("out.png","PNG")

And the resulting image:

There's certainly a lot of pretty detail there that we can explore!

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