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Need proof re prime of the form 8N +/1 
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#1
Nov1112, 09:02 PM

P: 894

I need help or direction on how to prove that if A = S^2  (T^2 + T)/2 Then 8A1 can not be factored into the form B*C where B and C are coprime and each of the form 8N+/3. For instance 4*81 = 33 can be factored as 3*(8+3) and 5*81 = 39 = 3*(8*23). Thus neither 4 or 5 can be expressed as S^2 (T^2+T)/2 where S and T are integers.
So far I have proven that if A = f(S,T) = S^2  (T^2+T)/2 then A = f(S',T') where S' = 3S + 2T +1 and T' = 4S + 3T + 1, but I don't know where to go from there. Any ideas. 


#2
Nov1412, 10:56 AM

P: 39

S^2 = 1, 4, 9, 16, 25 ...
(T^2 +T)/2 = 1, 3, 6, 10, 15, 21... 91 = 8; 8*81 = 63; 63 = 3*(8*33). therefore your statement is false. edit: nm, missed the co prime part. okay; i programmed a check up to many values, as far as i can tell this is true. how to prove ti is beyond me though. 


#3
Nov1412, 01:15 PM

P: 894




#4
Nov1412, 10:39 PM

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P: 9,871

Need proof re prime of the form 8N +/1
Suppose p is prime, 3 or 5 mod 8.
Easy to show that p cannot be expressed as 2*a^{2}b^{2}. Also seems to be true that if p2*a^{2}b^{2} then so does p^{2}. That looks like it might be associated with your observation. 


#5
Nov1512, 07:41 AM

P: 144

Hi Ramsey, your observation is a consequence of the following:
Lemma: Let N=2x^{2}y^{2} with x and y integers. Let pN be a prime of the form 8ką3. Then ord_{p}(N) is even. (By ord_{p}(N) we mean the exponent of p in the factorization of N.) Proof: First recall that 2 is a quadratic residue modulo a prime q if and only if q is of the form 8ką1. Since pN we have 2x^{2} = y^{2}(mod p). Since 2 is a quadratic nonresidue, it follows that y=x=0 (mod p), and all the terms of the equation N=2x^{2}y^{2} can be divided by p^{2}. Repeat as long as N has prime factors of the form 8ką3, and qed. Your observation follows immediately from this by setting N=8A1, x=2S, y=2T+1, and by observing that in a coprime factorization N=bc, all factors p^{a} are of the form 8ką1. I assume the lemma is well known, but I couldn't immediately find a reference. It is analogous to the celebrated theorem about sums of two squares, one version being: A positive integer N can be written as a sum of two squares if and only if for all primes p of the form 4k+3, ord_{p}(N) is even. 


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