Sum of Two SquaresDate: 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/ |
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