## JSH: z constraint and solving quadratic residues

I started a thread on solving quadratic residues mod p, which is about a simple technique for finding k, when

k^2 = q mod p.

Remarkably the thread has degenerated so I thought I'd make another one noting some about where the research result comes from, which is what I call the z constraint:

Given z^2 = y^2 + T

where T is composite, and y and z are coprime integers, for any factorizations of T into two positive integers f_1 and f_2, where there exists an odd prime p less than both, or if f_1 is the smaller factor p-f_1 is less than f_1, where also exists an alpha such that

k^2 = (1 + alpha^2)^{-1}(T) mod p

then

z = (1 + 2alpha^2)k/(2alpha)

in general when T mod 3 = 2, and for some reason I had with 50% probability when it doesn't though I'm not sure where I got that from. But when T mod 3 = 2, it is 100%.

So I just creatively used those equations with alpha=1, and T=2q mod p, to get:

Given a quadratic residue q modulo p where p is an odd prime and not 3, where you wish to find k, where

k^2 = q mod p.

where T cannot equal 2q, so you'd start with T = 2q+p or T = 2q - p,
and next you have to factor it:

With f_1 and f_2 where f_1*f_2 = T,

you get k from

k = 3^{-1}(f_1 + f_2) mod p.

Example: Let q=5, p=19, then T = 10 mod 19, and T = 29, works with f_1=29 and f_2 = 1, giving

3k = 30 mod 19, and k=10.

10^2 = 5 mod 19.

The original result follows from research of mine which is often ridiculed that I find it hard to even mention it.

Posters, of course, blame me. When to me it is bizarre how easily the math community can block ANY mathematical research idea. Just look at the nothing happening with my solution to binary quadratic Diophantine equations. No debates about correctness. But it doesn't matter to be correct.

They just slap a nasty label on you and proof doesn't matter and THEY don't give a damn about the importance mathematically of the result.

Don't tell me it's just a matter of putting it out there either. I've had these results for about a year now both posted on newsgroups and on my math blog.

So I'm talking old research.

Just do a search on "z constraint".

It's like what they did to me with Fermat's Last Theorem: they could get points just by ridiculing even the CONCEPT of working on the problem as if it automatically proved I was a crackpot and had to be wrong.

Then, dismissing actual research finds was academic.

Mathematicians can block public knowledge of mathematical discovery, at will.