Some of my history with modular

[JSH]

Back in 1996, wrote a paper, where now realize had a modular approach to a geometry problem. What I did was deliberately consider a unit in an imagined infinite space of spheres, where calculated the density within that unit for a packing, and noted that ANY perturbation lowered that density.

And that packing of course was hexagonal close, and I'd found a very basic and simple proof of the packing problem for spheres, and the paper was rejected by an editor at the Proceedings of the AMS as too simple. Which to me at the time was like, maybe? It has bothered me for over 20 years, and for a long time didn't talk about it much.

Do have a post talking a bit on my blog: http://somemath.blogspot.com/2008/05/spherical-packing-problem.html

Now though have LOTS of modular approaches, and with that abstraction in hand, yeah, it makes sense! Modular approaches greatly simplify but as an abstraction really came in with Gauss, though Euler did some work too.

Looks like after Gauss though, mathematicians ran away from them a lot, like consider arguments against modular in number theory with regard to factorization.

xy = P mod N

You can get a modular inverse for x or y in a simple way, which seems to tell you nothing. But consider instead from MY research, focus on a different equation gives a LOT more information.

x^2 - Dy^2 = F mod N

Looks like first posted close to this direction with: http://somemath.blogspot.com/2011/10/two-conics-equation-modular-solutions.html

That post is dated October 19, 2011. 

Talked doing it in this post without showing: http://somemath.blogspot.com/2012/06/quadratic-residue-modular-solution.html

Post dated June 25, 2012. Ok, found where FINALLY, had it. Post dated September 28, 2012:

http://somemath.blogspot.com/2012/09/generalizing-modular-solution.html

The equation x² - Dy² = F where all variables are non-zero integers can be solved modularly.

Find a non-zero integer N for which a residue m exists where: m² = D mod N, and r, any residue modulo N for which Fr^-1 mod N exists then:

2x = r + Fr^-1 mod N and 2y = m^-1(r - Fr^-1) mod N

if m does not share prime factors with N, otherwise: 2my = r - Fr^-1 mod N.