Talk:Fermat's theorem on sums of two squares

From Wikipedia, the free encyclopedia

This is the talk page for discussing improvements to the Fermat's theorem on sums of two squares article.
Please sign and date your posts by typing four tildes (~~~~). Put new text under old text. Click here to start a new topic. New to Wikipedia? Welcome! Ask questions, get answers. This is not a forum for general discussion about the article's subject.	Be polite Assume good faith No personal attacks Be welcoming	Article policies No original research Neutral point of view Verifiability

Contents 1 Don Zagier's proof 2 Thue's lemma ? 3 Self-Contradictory Article? 4 Converse

[edit] Don Zagier's proof

A recent addition to Don Zagier gives references to a short proof that perhaps deserves mention here: [1][2]. 165.189.91.148 20:43, 14 September 2006 (UTC)

[edit] Thue's lemma ?

There is no explanation as to why Thue's lemma redirects to this page. Can anyone help? DFH 18:52, 26 January 2007 (UTC)

[edit] Self-Contradictory Article?

"According to Ivan M. Niven, Albert Girard was the first to make the observation and Fermat was first to prove it."

"As was usual for claims made by Fermat, he did not provide a proof of this claim."

These two statements cannot both be accurate regarding a common subject, the incorrect one should be removed...

195.137.90.156 18:14, 15 June 2007 (UTC)

Fermat claimed he could prove it, but phrased it as a challenge to mathematicians (as was his usual practice). I will add "claim" to the second clause. Magidin 20:01, 16 June 2007 (UTC)

[edit] Converse

Suppose you have a number which is 1 mod 4 and you've written it as the sum of two squares. Does that imply primality? I suspect the answer is no but I'm not sure. Dcoetzee 02:33, 19 June 2007 (UTC)

No: any product of two numbers that can be written as a sum of two squares can itself be written as a sum of two squares, by the Brahmagupta-Fibonacci identity. As a consequence the integers that can be written as a sum of two squares are exactly those with the property that in their factorization into primes, any odd prime that is not congruent to 1 modulo 4 occurs to an even degree. If the number is odd, then it will be congruent to 1 modulo 4 necessarily. A very partial converse is that a positive integer can be represented as a sum of two squares with no common factors if and only if all its odd prime factors are of the form 4k + 1, except for the prime 2 which may occur to at most the first power.

The reason there is a focus on primes rather than general integers is the Brahmagupta-Fibonacci identity: once you know which primes can be represented, you will know exactly which numbers can be represented. Magidin 14:57, 19 June 2007 (UTC)