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Let $p=4k+1,$$p$ is a prime numbe,and $ k\in\mathbb{Z}$. Prove the existence of positive integer $a_1,a_2,\ldots,a_k$ and $b_1,b_2,\ldots,b_k$. $$ p=\frac{(a^{2}_{1}+1)(a^{2}_{2}+1)\cdots(a^{2}_{k}+1)}{(b_{1}^{2}+1)(b_{2}^{2}+1)\cdots(b_{k}^{2}+1)}. $$ My friend trying to prove the following for a long time,and I cann't solve it,I hope you can help me. I would be extremely happy if somebody could help me with this!

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    http://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares2012-04-24
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    *Let $n \in {\mathbb N}$, and $n > 3$. Prove the non-existence of positive integer $x, y, z$. $$x^n + y^n = z^n$$ My friend trying to prove the following for a long time,and I cann't solve it,I hope you can help me. I would be extremely happy if somebody could help me with this!* It is definitely not the question for stackexchange.2012-04-24
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    Where did the problem come from? Is there any reason to believe it holds? Are there any restrictions on $a_i$ and $b_i$? Context?2012-04-24
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    @pedja The question is not about the sum of two squares but about the complex fraction. Getting one from the other is not obvious, if the two things related at all.2012-04-24
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    @penartur: my first thought was to use results of sums of squares, and repeated application of the Brahmagupta-Fibonacci identity. What is (or isn't) obviously to you might not be obvious to others!2012-04-24
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    @penartur Why is this question *definitely not a question for stackexchange*? Where else would you go?2012-04-24
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    @draks Making it more obvious, would you consider a question "I want to write my Ph.D. diploma, i tried writing it for a long time but didn't succeed, please help me" as appropriate for stackexchange? Both this and OP questions are more like requests to do some heavy (and well-paid outside of the stackexchange) work/research for free. If one wants to prove some not-well-known theorem, they should do it by themselves or discuss it on some thematic boards / etc, stating the discussants as co-authors.2012-04-24
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    @draks Or here is another analogy: asking people to say how can you fix your broken ipad is the question; asking people to invent a new ipad for you (so that your company may became #1 on the tablets' market) is not the question, and asking it on a resource like stackexchange where the people help others for free is a bit inappropriate and impertinent.2012-04-24
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    @penartur (1) This site is for all math questions, simple or difficult. (2) Even if difficult questions were disallowed, generally there is no way to know in advance how difficult a solution will be. Teachers sometimes pose difficult (even unsolved) problems as exercises without any warning to students. Occasionally they are solved. Incorrect preconceptions about the difficulty of problems often times prevent one from discovering simple, beautiful solutions.2012-04-24
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    @Arturo Magidin:This problem comes from one of my friend,and I think it's right.I figured that the smaller the case.Thank you all the same.2012-04-24
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    @BillDubuque Still there is a difference between a difficult question and the work task.2012-04-25
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    @penartur:Thank you for your concern, I am a Chinese student, my English is terrible.I think I will try to overcome language difficulties.It will be a pleasant thing ,exchange Mathematics with you.thank you!2012-04-25

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