Recently I noticed a sum of squares result with integers, which is:
x2 + y2 = mn
will always have nonzero integer solutions for x and y, as long as m - 1 is a square and n is greater than 0.
For example: 1612 + 2402 = 174
Where I'm using a more elegant form than the more functional form I tend to show in posts when I'm talking about making it work, like how you calculate x and y.
Here though I want to talk about the mathematical foundations of the result, so it's worth it to show it at its most elegant as a basic fundamental result of number theory, which it is!
But from where does it follow?
Turns out there is a bit of history and the result builds on research of mine over more than a decade, which ironically resulted from my working at a problem so infamous in math circles I find I hesitate to mention it.
However, for a result this basic to have remained hidden for centuries, actually maybe a couple of thousand years, it follows there must be some new math involved. And there is, as I came up with an analytical approach I call tautological spaces, which was just extending modular algebra symbology.
That such an option was still available is extraordinary to me. But experienced math people can maybe see the difference with something like:
If x2 + y2 = z2 then (v2 - 1)z2 - 2xy = 0(mod x+y+vz), where v can be any value.
Having a modulus which is all symbols is a usual characteristic of this path of research. And you can see that result proven in a post where I step through a mathematical proof where its absolute truth is easy to show, where there is no room for logical doubt.
Using those techniques on a more complex equation I found a simpler way to generally reduce binary quadratic Diophantine equations. Which when used on the already reduced form of a general binary quadratic, gave what I now call the Binary Quadratic Diophantine iterator.
And I cover some of those foundations in mathematical detail in a prior post:
This mathematical approach of using more symbols in the modulus with modular algebra is the foundation, so links back really to Gauss and his work laying the foundations of modular arithmetic, so I like to say it's just a natural extension from there.
It's incredibly powerful as an approach, which tends to give what I call infinity results, as modular algebra kind of can't help but cover infinity.
Having scratched the surface and looked for easy answers, I got a simpler more general way to reduce binary quadratic Diophantine equations, new ways to calculate the number of quadratic residue pairs, and a way to explain the fundamental solution of x2 - Dy2 = 1, among other things, like my result above!
And that was just looking for the easy things.
You have to choose discovery.