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

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Bruno
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PostSubject: [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
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Mohammad
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PostSubject: 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 Crying or Very sad
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.
Crying or Very sad
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Bruno
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PostSubject: 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). cheers
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Mohammad
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PostSubject: 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. bounce
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PostSubject: 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. bounce

Yes, there is probably another proof!
But the algebraic method is elegant. I love you
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