Need proof re prime of the form 8N +/-1

[ramsey2879]

I need help or direction on how to prove that if A = S^2 - (T^2 + T)/2 Then 8A-1 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*8-1 = -33 can be factored as -3*(8+3) and 5*8-1 = 39 = 3*(8*2-3). 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.

 

[phillip1882]

S^2 = 1, 4, 9, 16, 25 ...
(T^2 +T)/2 = 1, 3, 6, 10, 15, 21...

9-1 = 8; 8*8-1 = 63; 63 = 3*(8*3-3).
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.

 

[ramsey2879]

S^2 = 1, 4, 9, 16, 25 ...
(T^2 +T)/2 = 1, 3, 6, 10, 15, 21...

9-1 = 8; 8*8-1 = 63; 63 = 3*(8*3-3).
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.

Thankyou for your post. Glad to see that someone was interested enough to check my apparent finding.

 

[haruspex]

Suppose p is prime, 3 or 5 mod 8.
Easy to show that p cannot be expressed as 2*a²-b².
Also seems to be true that if p|2*a²-b² then so does p². That looks like it might be associated with your observation.

 

[Norwegian]

Hi Ramsey, your observation is a consequence of the following:

Lemma: Let N=2x²-y² with x and y integers. Let p|N be a prime of the form 8k±3. 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 8k±1. Since p|N we have 2x² = y²(mod p). Since 2 is a quadratic nonresidue, it follows that y=x=0 (mod p), and all the terms of the equation N=2x²-y² can be divided by p². Repeat as long as N has prime factors of the form 8k±3, and qed.

Your observation follows immediately from this by setting N=8A-1, x=2S, y=2T+1, and by observing that in a coprime factorization N=bc, all factors p^a are of the form 8k±1.

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.