Years ago discovered there were these powerful control equations that force things like distribution of quadratic residues. Learning that happened at some point where was long enough ago not sure when, but for sure one of the important revelations was found fortuitously from staring at:
(n - m)2 = m2 + 1 mod D-1
which follows from:
(n-m)2 - Dm2 = 1
Should note don't use ≡ because I use modular algebra SO much and why add some extra line under equals all over the place? Seems like useless work to me. Oh yeah, to try and help those who might need the info wrote a post explaining mod years ago.
So was pondering a slight variation from the usual x2 - Dy2 = 1. And had been playing around with the different form and became fascinated by the forced quadratic residue there.
Very glad I noticed it!
Which lead me to finding out about quadratic residue pairs. Which I'd never heard of, until started searching to see if I'd found something new. So there, yes existence had been noticed before. Ok. Is cool to learn more historical number theory to research your own discovery. But then also I figured out how to prove some things, which was new.
You can kind of see things from usual form as well:
x2 = y2 + 1 mod D-1
And you can easily solve for unknowns modulo D-1 with either form.
Using my alternate: 2m = n-1(n2 - 1) mod D-1
Which you need for a proof, which is how to derive the count of quadratic residue pairs which for a long time was one of the most popular posts on this blog. And yeah can use it that way as IS one of the most powerful influences across integers, and yeah can use it to count. It is one of the infinity tools of number theory, which helps rule over quadratic residues.
Prior explanations for the count I found, using web search, relied on the pigeonhole principle. They had no clue.
For people into number theory, you can just go check! Web search. And is how I found much so I know information in this area is readily available. That reference reality is a major plus of our times.
So far as I know I gave the first direct derivation which not only lets you know why, it allows you to do more. Will admit am more into the 'why' as exciting, so just talked more you can do in my post as can expand to counting in more situations! But didn't do myself. But is a good example to see how a derivation can give more.
Especially is that odd feeling when you notice also an easy path that somehow escaped others. Why not figured out a century ago? Who knows. But probably because Gauss pioneered modular in his time, and I picked up where he left off, in ours.
Is odd too. Looks like in past, mathematicians knew more of modular arithmetic, but didn't realize a full modular algebra, which I did. And for me to get there had to create an entire mathematical discipline and pioneer abstract reductionism so maybe is harder than I wish to accept!
For me? Is just what I did. Guess that doesn't give me the perspective of looking at from a distance.
The joy of discovery I guess can be hard to describe but it definitely can motivate to find more, which I think is great. There is that thrill, for what YOU find, with your mind, and mental effort. Is a thrill...hard to describe.
Here again my modular approach leads to a short, powerful answer and in this case, a first derivation.
And it dawns on me that modular as a first reach in problem solving may be something very new to the human species, but why?