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# [SOLVED] Primes of the form 8n+1, 8n+3

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 Subject: [SOLVED] Primes of the form 8n+1, 8n+3   Mon Nov 09, 2009 1:10 am Show that every prime of the form 8n+1 or 8n+3 can be written as a^2+2b^2 for a, b positive integers.(My estimate of the difficulty : 8/10)Last edited by Bruno on Fri Nov 13, 2009 4:32 pm; edited 1 time in total

Descartes

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 Subject: Re: [SOLVED] Primes of the form 8n+1, 8n+3   Thu Nov 12, 2009 11:22 pm I am too old to remember some stuff I just know by wonderful Gauss's theorem of quadratic reciprocity it is solvable in mod p. Anyway, I'd love to see the proof and then I will cry why I am far away from my glorious days.

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 Subject: Re: [SOLVED] Primes of the form 8n+1, 8n+3   Fri Nov 13, 2009 4:30 pm It is Fermat who conjectured this result.The first step is to show that -2 is a quadratic residue for primes of the form 8n+1 and 8n+3. (In the case of 8n+1, both 2 and -1 are quadratic residues, so -2 is also; in the case of 8n+3, neither 2 nor -1 are quadratic residues, so -2 is.)Thus we can find integers m, k such that m^2+2=kp. Thus (m+√-2)(m-√-2)=kp. Since p divides the left side, but neither of the factors on the right, and since Z[√-2] is a unique factorization domain, p must factor in Z[√-2]. Thus we can write p=(a+b√-2)(c+d√-2). Taking norms on both sides we have p^2=(a^2+2b^2)(c^2+2d^2), so p=a^2+2b^2=c^2+2d^2.(Using the fact that at most two prime ideals in an quadratic integral extension R/Z can lie over a prime ideal in Z, we can also deduce that the representation of p as a^2+2b^2 is unique).

Descartes

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 Subject: Re: [SOLVED] Primes of the form 8n+1, 8n+3   Fri Nov 13, 2009 8:33 pm oh, I have never studied number theory from this perspective, but there should be another non-machinery proof.

Posts : 184
Join date : 2009-09-15
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 Subject: Re: [SOLVED] Primes of the form 8n+1, 8n+3   Sat Nov 14, 2009 4:13 am Mohammad wrote:oh, I have never studied number theory from this perspective, but there should be another non-machinery proof. Yes, there is probably another proof!But the algebraic method is elegant.

 Subject: Re: [SOLVED] Primes of the form 8n+1, 8n+3

 [SOLVED] Primes of the form 8n+1, 8n+3
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