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12/06/2003 at 11:52PM PST, ID: 20818639
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7.6

x^2 + 2y^2

Asked by lj8866 in Math & Science

Tags: descent, fermat, method

I've been reading up on number theory and sum of two squares, and there's basically a theorem that states that if a prime p is congruent to 1 mod 4, it can be expressed as the sum of two squares (x^2 + y^2). This is proven using Fermat's method of descent andcan be found in any textbook. Now the textbook I am using had a note that numbers of the form x^2 + my^2 are also similarly proven. I tried to look at primes of the form x^2 + 2y^2 as the simplest case. After some experimentation, I found the pattern p can be expressed as x^2 + 2y^2 if p = 1,3 (mod 8) and not if p = 5,7 (mod 8). Now I don't know how to prove this though, in the way that they say _similarly_. Any ideas?

Also, the textbook claims that the fact that p = 1 mod 4 implies p = x^2 + y^2 also means x and y are unique (if we consider positive integers). I tried this out a few times but I also don't know how this is true. Can anyone show me why this is so? Thanks.

I'll try to split up points for each question.

-lj
[+][-]12/06/03 11:54 PM, ID: 9891400

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[+][-]12/07/03 05:33 PM, ID: 9893957

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[+][-]12/29/03 06:51 AM, ID: 10010746

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Zone: Math & Science
Tags: descent, fermat, method
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Solution Provided By: relf
Participating Experts: 2
Solution Grade: A
 
 
[+][-]04/28/04 11:11 AM, ID: 10942066

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[+][-]05/02/04 09:16 AM, ID: 10972081

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