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

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.