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I have received the book "Proofs from the books" 4th edition by Springer, and I found 3 problems on Number Theory relate to L.Euler. They are "Representation number as sum of two squares", "Quadratic Reciprocity" and "Three times $\dfrac{\pi^2}{6}$".

When I watched the public lecture of Clay Institute: Clay Public Lecture-A Tribute to Euler,

I saw the problem of partitioning number, say briefly $O_{n}=D_{n}$ in which $O_n$ is the number of ways to write $n$ as sum of odd number that small than $n$, $D_{n}$ is the number of ways writing $n$ as sum of different number that all small than $n. Professor Dunham presented a very nice proof, also related to Euler(the video lecture is available on that web, so you should download and watch it).

So, there are two question. Do you think this proof should be included in the book "Proofs from the book" ?

Are there any other beautiful proofs you think that are not mentioned in "The Book" ? Paul Erdos has gone to Paradise, but I think we should continue his work, completing "The Book".

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closed as not constructive by lhf, Hans Lundmark, jspecter, Asaf Karagila, Willie Wong Nov 3 '11 at 8:58

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

    
I see 4 votes to close this question as "not a real question". Would the last one who votes to close please leave an explanation as to why s/he did choose to do so? The question if that specific argument should be included in "The Book" is probably unanswerable. However, I see nothing wrong with creating a big list of clever and beautiful proofs (as long as they're not just reproductions from the book by Aigner-Ziegler). I don't remember having seen such a thread here. –  t.b. Nov 3 '11 at 8:35
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I voted to close, not as "not a real question", but as "not constructive" because the question is too subjective and discussion-y. Beauty is in the eye of the beholder; for example, I would consider a 10 page proof that while long, illustrates precisely the ideas behind the proof more beautiful than a 3 sentence "clever" proof that obscures all the meaning. And I know quite a few people who would think the reverse is true. I am happy with big-list questions. I am not happy with big-list questions where the answer can be practically anything. –  Willie Wong Nov 3 '11 at 9:02
    
If the OP or @t.b. can provide a reasonable set of criteria of what it means for a proof to be "clever" or "beautiful", I'll be willing to reconsider my position. (I don't even ask for a definition; nor do I ask for a set of criteria that I agree with. I ask for a reasonably objective [objective because this would be a Wiki effort; I have no qualms about Aigner and Ziegler coming up with their private subjective criteria and writing a book with examples illustrating them] list of features that we can tick off and say: this proof belongs in the list.) –  Willie Wong Nov 3 '11 at 9:09
    
@Willie: I totally agree with you. I mainly did not vote to close because I would have had to close this question as "not a real question" and couldn't have justified it (I would have gone for "subjective and argumentative/not constructive" had I had the option). I cannot and do not want to try to come up with a definition or criteria, and suspect that whatever I'd like to see is essentially covered by that book, anyway, so I don't insist that this question needs to be reopened. –  t.b. Nov 3 '11 at 9:15
    
I think there is a misunderstanding here. I means, do you think there are some other beautiful proofs of some beautiful problems those are not mentioned in the book ''Proof from the book''? This is really a soft question, so please do not take it in a too serious way. By beautiful proof, I mean a elegant, clear in the idea proof, I do not think a tricky proof is beautiful. –  Nguyễn Duy Khánh Nov 3 '11 at 10:03

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