Talk:Fermat's theorem on sums of two squares

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)

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)

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)

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)

3/26/2012 version reads, I think incorrectly: Fermat usually did not prove his claims and he did not provide a proof of this statement

Like Archimedes and others, Fermat often didn't publish his proofs, that doesn't mean he didn't prove it. I think Fermat did prove this result, by infinite descent. For another example of a scientist being secretive about his results, see Hooke's Law.198.189.194.129 (talk) 18:57, 26 March 2012 (UTC)

By "Fermat usually did not prove his claims" it is meant that he did not offer or provide any proofs of the claims made. That statement is correct. What your beliefs (or mine for that matter) are about the matter are irrelevant. Magidin (talk) 20:14, 26 March 2012 (UTC)

Yes, but I believe that reputable published sources say that he usually proved his theorems. I will try to find such a source.--Rich Peterson198.189.194.129 (talk) 18:34, 27 March 2012 (UTC)

Oh, I forgot to say that if we don't know one way or another if he proved his claims, but he did not publish proofs, it would be better to say {"He did not publish his proofs, if any."} which avoids a possible inaccuracy.Best regards, richard Peterson198.189.194.129 (talk) 18:39, 27 March 2012 (UTC)

I am not familiar with any source that says he provided proofs of his theorems as a matter of course or regularly. For the sum of two squares, Stillwell writes in Theory of Algebraic Numbers (page 12) "Fermat thought he could prove these theorems, and he was probably right, as proofs eventually published by Euler were based on Fermat's method of infinite descent." This indicates that Fermat never offered proofs of these results. Edwards, in Fermat's Last Theorem: A genetic introduction to number theory writes in page 16 about the theorem here in question: "Fermat stated [that every prime of the form 4n+1 can be written as a sum of two squares] many times and stated very definitely that he could prove it rigorously, although as usual he is not known ever to have put the proof in writing." [emphasis added] His proof of Fermat's Last Theorem for the $n=4$ case is singular because it was written down. Again, Edwards writes (page 10): "It seems that only one proof is to be found in all of Fermat's surviving work on number theory. It is his proof of a particular proposition which Fermat stated a number of times in his correspondence but which, characteristically, he did not prove in his correspondence, leaving it to his correspondents to try to solve the problem for themselves. The proof, like the statement of the Last Theorem, was found by his son Samuel in the margin of his copy of Diophantus and was included in the posthumously published works as Observation 45 on Diophantus." [emphasis added]. Again, you seem to be reading "did not prove/did not provide proof" as "he, personally, did not prove them/could not prove them." The statement is about what he published, is known to have put in writing, or circulated; not what he may have done in private or in the inside of his head. Magidin (talk) 20:30, 27 March 2012 (UTC)

I think you're correct about the sources then. But still, you said above {By "Fermat usually did not prove his claims" it is meant that he did not offer or provide any proofs of the claims made.} It seems to me better to change the article to: "Fermat usually did not offer or provide any proofs of the claims made." Of course, that's basically the last few words of your sentence.198.189.194.129 (talk) 18:12, 28 March 2012 (UTC)

False statement attributed to Albert Girard

The first sentence in the "History" section begins: "Albert Girard was the first to make the observation, describing all positive integer numbers (not necessarily primes) expressible as the sum of two squares of positive integers ...". However, it is not true that all positive integers are expressible as the sum of two squares of positive integers. Counterexamples include 1, 3, 6, 7. Mksword (talk) 16:37, 14 October 2020 (UTC)

This sentence doesn't say that Girard asserted all integers are sums of two primes, it just says Girard was the first to describe all the integers that are the sums of two primes. --Sapphorain (talk) 17:10, 14 October 2020 (UTC)

Thank you, Sapphorain; I see that I made a mistake. However, I still find the sentence to be unsatisfactory, because it says that Girard "described" those numbers but it does not indicate what description he gave. Do you know what description Girard gave of those numbers? Mksword (talk) 07:30, 22 November 2020 (UTC)

Yes. See this.--Sapphorain (talk) 20:12, 22 November 2020 (UTC)

Thank you again, Sapphorain. I did see that in the article's footnote #2. I don't read French. Is it available in English? Mksword (talk) 02:49, 24 November 2020 (UTC)

E. Dickson mentions this description thus: "A. Girard ... had already made a determination of the numbers expressible as a sum of two integral squares: every square, every prime 4n+1, a product formed of such numbers, and the double of the foregoing". (History of the theory of numbers, Vol. 2, p.227).--Sapphorain (talk) 08:28, 24 November 2020 (UTC)

Muchas gracias, Sapphorain. I incorporated Dickson's quote into the article's footnote #3. Mksword (talk) 08:13, 25 November 2020 (UTC)

Merging the proofs in this article

The content of this article is reduced to the statement of the theorem, its history, an algorithm, and a short section on related results. For such a short article on a theorem, there is no reason to split the proofs in a separate article, specially when, as in this case, the main interest of the theorem is the number of the proofs, and their difficulties for an apparently simple result.

So, I'll merge here Proofs of Fermat's theorem on sums of two squares. D.Lazard (talk) 15:53, 21 July 2021 (UTC)

easy to disprove

counter examples

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, ... (OEIS: A002145)

These are the primes of the form 4k + 3. If the number has a prime factor of the form (4k + 3)ⁿ for an odd n (which is 1 for 4k + 3), then a² + b² is impossible. 94.31.85.138 (talk) 21:02, 13 November 2023 (UTC)

What is it you believe you are "disproving", exactly? Magidin (talk) 22:06, 13 November 2023 (UTC)

There are many primes that are not a sum of 2 squares, while on the collatz conjecture for example, all the tested numbers so far eventually go to 1. 94.31.80.138 (talk) 05:43, 14 November 2023 (UTC)

The theorem says that an odd prime is the sum of two squares if and only if it is congruent to 1 modulo 4. So I ask again: what is it you believe you are "disproving", by giving examples of primes that are not the sum of two square and not congruent to 1 modulo 4? All those examples agree with the theorem. And the collatz conjecture has nothing to do with this, which suggests that you are either a crank or don't know what you are talking about (or both). 18:34, 14 November 2023 (UTC)