Sum of Two Squares

Date: 05/26/2003 at 02:40:31
From: David
Subject: Sum of Two Squares

I was wondering if there was some way to generate the squares used to 
sum up to a particular number. For example, in the archive I found 
that the least number expressible as the sum of two squares in 12 
different ways is 160225, and its breakdown is:

   160225 = 400^2 + 15^2,
          = 399^2 + 32^2,
          = 393^2 + 76^2,
          = 392^2 + 81^2,
          = 384^2 + 113^2,
          = 375^2 + 140^2,
          = 360^2 + 175^2,
          = 356^2 + 183^2,
          = 337^2 + 216^2,
          = 329^2 + 228^2,
          = 311^2 + 252^2,
          = 300^2 + 265^2.

So, I was wondering if you knew of some way to generate the sequence 
[400, 399, 393, 392, 384, 375, 360, 356, 337, 329, 311, 300].

Date: 05/26/2003 at 08:11:13
From: Doctor Schwa
Subject: Re: Sum of Two Squares

What you need to do is factorize the number into COMPLEX primes, and 
then look at all the ways of combining them.

For instance, suppose you start with 65 = 5 * 13. Then you continue 
factoring into

   65 = (2+i)(2-i)(3+2i)(3-2i).

Then, if you write

   65 = [(2+i)(3+2i)] [(2-i)(3-2i)]

you get

   65 = [4 + 7i][4 - 7i]

so 65 = 4^2 + 7^2,

but if you write

   65 = [(2+i)(3-2i)] [(2-i)(3+2i)]

you get

   65 = [8 - i] [8 + i]

so 65 = 8^2 + 1^2.

Does that help?

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/