Talk:Proofs of Fermat's theorem on sums of two squares

message: A recent addition to Don Zagier gives references to a short proof that perhaps deserves mention here:  . 165.189.91.148 20:43, 14 September 2006 (UTC)
thread_title:  Don Zagier's proof 

message: Two more proofs can be seen here:
http://planetmath.org/encyclopedia/ProofOfThuesLemma.html
(I don't know why they call the result Thue's Lemma.)
:Feel free to add an outline of those proofs, or to add the reference. Magidin 15:08, 21 February 2006 (UTC)
thread_title:  Other proofs 

message: It seems misleading to say "Since Z[i] is a unique factorization domain (in fact, a Euclidean domain), every ideal is principal."
Z[i] is a Euclidean domain and as such it is a PID, but this is not guaranteed by the fact that it is a unique factorisation domain.
Suggest the following revision: "Since Z[i] is a principal ideal domain (in fact, a Euclidean domain), every ideal is principal." Or, alternatively, just say that this follows from the fact that Z[i] is a Euclidean domain. —The preceding unsigned comment was added by 163.1.146.236 (talk) 18:27, 1 February 2007 (UTC)
replies: text: I made the change, but I wanted to point out the reason it was written as it was: basically, it was paraphrased from Dedekind's argument. For rings of integers of number fields, the ccondition of being a PID and being a UFD are equivalent; at the time, these were the only general "rings" of interest. Dedekind was interested n unique factorization, not the PID property, though he had shown that the two were equivalent. Magidin 19:57, 3 February 2007 (UTC)
thread_title:  Unique Factorisation Domain 

message: In Euler's proof, in his second statement, it is not necessary to state that p2 +q2 divides (ap+bq)2, because in the final expression, a fraction is left out. But a better statement would be that the lift hand side of the final expression is an integer and the second term in the right hand side is an integer, so the first term better be an integer.
This better be because, it does not mean that (p2 + q2)2 divides (ap+bq)2 from the above statement
:Whether this is better or not is not really material. The proof is as it was quoted in Edwards's book, and is meant to be the proof that was actually given by Euler. No doubt it can be improved, but that is not the point here. Also, please remember to sign and date your comments. Magidin 18:24, 25 March 2007 (UTC)
replies: text: "If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares."
Now 45 = 36+9 = 62+32, so 45 is a sum of squares. And 5 = 12+22, so 5 is a prime which is a sum of squares. But 45/5 = 9, and 9 is not the sum of squares.
I don't see a quick and easy edit to fix the problem
and don't have access to an authoritative text
of Euler's proof. Anyone know what's wrong? --Michael Ross (talk) 01:37, 3 October 2010 (UTC)

text: 9 is the sum of the square of 3 and the square of 0. Every square is the sum of two squares. Magidin (talk) 01:49, 3 October 2010 (UTC)

text: Perhaps you can say explicitly what you want to add as a preamble to the paragraph, and where the addition would be. It would be easier to discuss specifics. Note that because this is in a sense a subsidiary page focusing on proofs, a certain amount of mathematical maturity and knowledge is assumed; the main page is the one supposed to be aimed at the members of the general public, with this page being referenced there as one providing more mathematical detail and necessarily assuming more mathematical understanding from the reader). Magidin (talk) 04:11, 19 August 2015 (UTC)
thread_title:  Euler's proof 

message: "Lagrange proved that all forms of discriminant − 1 are equivalent."  This can't be right.  (x2 + y2) and (- x2 - y2) represent different sets of integers, nonnegative and nonpositive, respectively, so they are not equivalent.  Should this read "Lagrange proved that all forms of discriminant -1 with a>0 are equivalent"? 68.162.145.63 (talk) 12:50, 11 March 2009 (UTC)
replies: text: Hrmph. I took that out of Stillwell's introduction to Dedekind's Theory of Algebraic Integers, and there is no such restriction.He seems to be fudging over forms and reduced forms: that Lagrange showed any form with negative discriminant can be transformed into an equivalent form which is reduced in the sense that |b|\leq a\leq c (restriction to reduced form occurs just before and just after stating that forms of discriminant -1 are all equivalent). But even taking that into account, it is clear there is something getting lost somewhere... Magidin (talk) 13:28, 13 March 2009 (UTC)
replies: text: I really believe that in the meanwhile that section should be deleted, or maybe there should be an alert that warns non-suspicious users like me from following that statement. Peleg (talk) 18:27, 10 April 2009 (UTC)
replies: text: Deleting the whole section is incredibly draconian. I added the restriction to reduced forms for now. Magidin (talk) 01:31, 11 April 2009 (UTC)
thread_title:  Lagrange's proof through quadratic forms 

message: I've just reworded the proofs by Dedekind. One notable point is that I dropped the mention of "degree of a prime ideal" in the first proof, because it plays no role, and because I have no idea what it meant in this context. But maybe somebody would like to explain this anyway? Also it might be worth while to note that the second proof actually provides an efficient method to find x and y, provided an integer m is known (which is easy, just try \tfrac{p-1}4-th powers of random integers, each of which has a 50% chance of being such an m): compute the gcd of p and m+i in the Gaussian integers. Or maybe this should go in the parent article? Marc van Leeuwen (talk) 12:43, 5 March 2011 (UTC)
thread_title:  Degree of a prime ideal? 

message: This article is completely remiss in not providing an example of any kind to demonstrate the practical application of the proof - that is, not the proof itself, but just what the theorem shows.
129.63.2.71 (talk) 10:09, 17 July 2014 (UTC)
replies: text: This article is about the proofs; the main page contains examples of what the theorem says. Magidin (talk) 16:32, 17 July 2014 (UTC)
thread_title: Before you explain the theorem, illustrate what the theorem means!