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m2012
 Post subject: primes of the form 4k+1
PostPosted: Sat, 2 Jun 2012 15:44:42 UTC 
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Prove that if p is a prime number of the form 4k+1 then there are integers x, y such that x^2+1=py^2.
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outermeasure
 Post subject: Re: primes of the form 4k+1
PostPosted: Sat, 2 Jun 2012 15:54:50 UTC 
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m2012 wrote:
Prove that if p is a prime number of the form 4k+1 then there are integers x, y such that x^2+1=py^2.


Units of \mathcal{O}_{\mathbb{Q}(\sqrt{p})}.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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m2012
 Post subject: Re: primes of the form 4k+1
PostPosted: Sat, 2 Jun 2012 16:01:06 UTC 
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Sorry but i don't understand your answer :(
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Shadow
 Post subject: Re: primes of the form 4k+1
PostPosted: Sat, 2 Jun 2012 20:46:50 UTC 
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m2012 wrote:
Sorry but i don't understand your answer :(


Then perhaps you should explain your background and what you have tried on this problem so we know what level of answer is appropriate for your abilities.

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m2012
 Post subject: Re: primes of the form 4k+1
PostPosted: Sun, 3 Jun 2012 10:43:04 UTC 
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Would you please show me the idea of elementary proof, without quadratic integer rings theory or Pell's equatioion. I know congruences theory, for example :)
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