Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 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 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 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. 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 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). 

Mohammad Descartes
Posts : 100 Join date : 20091105 Age : 34 Location : Right behind you
 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 nonmachinery proof. 

Bruno Admin
Posts : 184 Join date : 20090915 Age : 31 Location : the infinite, frictionless plane of uniform density
 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 nonmachinery proof.
Yes, there is probably another proof! But the algebraic method is elegant. 

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