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Let $F = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $F$. Let $m$ be an integer. If $m = ax^2 + bxy + cy^2$ has a solution in $\mathbb{Z}^2$, we say $m$ is represented by $F$. If $m = ax^2 + bxy + cy^2$ has a solution $(s, t)$ such that gcd$(s, t) = 1$, we say $m$ is properly represented by $F$.

My question Is there any other proof of the following theorem other than the Gauss's original proof? Since this theorem is important, I think having different proofs is meaningful.

It would be also nice if some one would post a modern form of the Gauss's proof, since not everybody can have an easy access to the book.

Theorem(Gauss: Disquisitiones Arithmeticae, art.154) Let $ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. Let $D$ be its discriminant. Let $m$ be an integer. Suppose $m$ is properly represented by $ax^2 + bxy + cy^2$. Then $D$ is a quadratic residue modulo $4m$.

EDIT The Gauss's DA is notorious for its difficult read. This was even so for his contemporaries. Dirichlet devoted a lot of time to simplify DA. There is a legend that Dirichlet always carried DA in his travels. Gauss's proof often uses a "magic" equation which seems to come out of nowhere. One of the reasons is that, as he wrote, he could not afford elaborate proofs due to lack of enough available pages for an economical reason. So I think it would be nice if there is a more natural proof.

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    Doesn't he prove this in his book?2012-09-04
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    @PeterTamaroff You know there are usually several different correct proofs of a mathematical theorem.2012-09-04
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    You should be saying what kind of proof you want. I'm reading the proof just now.2012-09-04
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    Gauss's proof of art.154 in my copy seems to be very clear and concise - what problem do you have with his proof, exactly?2012-09-04
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    @JohnSenior I think having a different proof doesn't hurt.2012-09-04
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    Yes, But you haven't answered my question: what problem do you have with his clear and concise proof?2012-09-04
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    @PeterTamaroff A different proof. If there are several different proofs, that would be nice.2012-09-04
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    @JohnSenior Do I have to have a problem with his proof in searching for other proofs?2012-09-04
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    But do you want what: a more contemporary proof? Shorter? Clearer? More convoluted? Using other theory? Avoiding certain axioms? Let's put down some variables!2012-09-04
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    It might be nice if you actually answered a question.2012-09-04
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    @PeterTamaroff Any correct different proof will do.2012-09-04
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    @JohnSenior Since the theorem is important, I think it's natural to looking for the different proofs.2012-09-04
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    Still no answer - I sign off2012-09-04
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    I have no idea why this question was downvoted. I don't think asking different proofs should be frowned upon.2012-09-04
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    What is frowned upon is that you don't give any hint of what you really want. I don't think there are many people that are going to read this and go "OK, let's find *some* proof, using *some* theory, that Makoto will like." Most seem to be cool with Gauss' proof. Maybe you can answer what we ask you so you can get what you want. *"But do you want what: a more contemporary proof? Shorter? Clearer? More convoluted? Using other theory? Avoiding certain axioms? Let's put down some variables!"*2012-09-04
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    @PeterTamaroff I said any correct proof will do.2012-09-04
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    @MakotoKato But not Gauss'?2012-09-04
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    @PeterTamaroff Please read my question.2012-09-04
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    OK, I can do something for you then.2012-09-05
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    Someone should ask a meta question about how to ask for alternative proofs of results with well-known or readily available proofs.2012-09-05
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    @William Why don't you do that if you think so?2012-09-05
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    I confess I am greatly saddened by the 6 downvotes to this question. I cannot imagine why anyone would downvote it. Please be more constructive. If you think the question needs improvement then please say why and/or help to improve it.2012-09-05
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    I understand that MSE, MO, and many other venues are viewed as opportunities for "competition", adversarial attitudes, and so on, but I wish it were otherwise. That is, I would wish that questions would not be asked as "challenges", but as requests for information. I am oh-so-well aware of the competition element in mathematics... but this is of no consequence to furthering it.2012-09-05
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    @Peter Many of the proofs in Gauss's *Disq. Arith.* are very old-fashioned or far from optimal; nowadays they can be presented much more slickly and with greater insight. So it is quite natural to pose a question such as that above. Further, comparing old proofs to new proofs often has great pedagogical value.2012-09-05
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    @JohnSenior Please see my prior comment.2012-09-05
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    @paulgarrett I see nothing at all "challenging" in MK's question. But, alas, I cannot say the same for some of the comments.2012-09-05
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    @BillDubuque I concurr with you. I upvoted this, and I'm really trying to help. I hope I'm coming off as that, and not otherwise. Would you give a hand with the mysterious product Gauss introduces? I recall you do some of that moves in your proofs too!2012-09-05
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    @JohnSenior The Gauss's proof is correct. I have no problem with it except that I have no idea how he came up with the crucial equation of his proof.2012-09-05
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    @PeterTamaroff it is Gauss composition, which takes two forms of the same discriminant and gives a third. It respects "equivalence." On the level of equivalence classes of forms, it makes a group. The difficult part of that proof is associativity. Dirichlet made a big simplification with his "united" pairs of forms. Many modern books do Gauss composition, some better than others. I like Buell. For one thing he does full detail on positive forms and indefinite forms in the same volume.2012-09-05
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    I'm also upvoting this question, more as a protest for the easy downvotes than anything else. It is always nice to have several proofs of some given result , e.g. see the galore of proofs of $\,\zeta(2)=\pi^2/6\,$ .2012-09-05
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    @WillJagy Interesting!2012-09-05
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    I voted to close this question. Any request for different proofs, etc., should include some *motivation*. Saying "not everybody can have an access to the book" is not motivation at all, and has never been a part of what math.SE is about.2012-09-05
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    @SteveD "Any request for different proofs, etc., should include some motivation." Could you explain why this is so? Anyway, I wrote that the theorem is important. That's one of my motivations.2012-09-05
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    @MakotoKato: it is like if I asked a question, "reading my linear algebra book, it said every vector space has a basis. Can I see different proofs of this result?". *Why* do you want different proofs? If questions that simply say "need different proofs of X" were allowed, the site would be inundated with such requests.2012-09-05
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    @SteveD My question is not of that sort. I took for granted that my question should be of interest for anyone who has the basic knowledge of elementary number theory.2012-09-05
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    @MakotoKato: fair enough. But you asked for different proofs, with the original reason being that not everyone had access to the book. Indeed, the highest voted answer as of now is simply *the exact proof* from the book you mentioned. I don't see why this question needed to be asked in the first place, but I will agree it has evolved into something worthwhile via the comments and subsequent questions.2012-09-05
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    @SteveD Do you still think this question should be closed? If yes, why?2012-09-07
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    @JohnSenior I answered your question. So I think there is no reason for you to sign off unless you have no problem with my question.2012-09-07
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    @SteveD Did you reset your vote for close?2012-09-08
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    I certainly did not. Again, it was the great community here and their thoughtful and insightful comments that made this page worth visiting. The question had nothing to do with it.2012-09-08
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    @SteveD You said the lack of the motivation was the reason for the vote to close. I wrote the motivation.2012-09-08
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    @SteveD I'm waiting for your reply. This is important for this thread.2012-09-08
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    It is not important for this thread at all, where are you getting that? Please stop pinging me, my interest in this thread has waned.2012-09-09
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    @SteveD You voted to close this thread. It is obviously important to this thread(it takes only 5 votes to close the thread). I'm surprised you don't recognize this. Please use your privilege with discretion. If the reason for the vote is no longer applicable to the question, please reset the vote.2012-09-09
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    It is a privilege I have earned, and it not your right to question what I do with it, just as I don't question why you ask just so many questions! I voted to close this question because it was worthless, and now by the magic of the community, has turned into something worthwhile. That has **nothing** to do with your original question. In fact, as activity has faded, I would indeed like to see this question closed. Even more so because you can't stop responding to me after I've clearly asked you to do so.2012-09-09
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    @SteveD I asked proofs more natural than the Gauss's original proofs because his proof seems to come out of nowhere. Please explain why my question is worthless.2012-09-09

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