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### 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/
```
Associated Topics:
College Imaginary/Complex Numbers
College Number Theory
High School Imaginary/Complex Numbers
High School Number Theory

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