Great advances in mathematics can lead to stunning simplifications. Which definitely makes easier to check. But can also fascinate as wonder about how our species learns.
Like with integers have quite a few results that show that key to their behavior is a single equation: x2 - Dy2 = F
I like to call it the two conics equation. As dependent on D, either gives hyperbolas or ellipses.
But even more important than the explicit form is the modular: x2 - Dy2 = F mod N
My most recent result MAY show how given one quadratic residue m modulo a composite, you can easily find another, by leveraging the power of the modular inverse. But haven't tested it, as is my hottest result yet, in many ways. But also is just a few days old where can take me YEARS to thoroughly test.
While will be a year come May 5th first posted then on May 9th more objectively talked a path from that modular form, to a new primary way to solve for the modular inverse. Feel like have talked it better with post:
My modular inverse method
That was made in December 2017. Does take time to process these things, where for me lots is emotional, which exasperates me somewhat. So I like to remind: math and emotion? Do not mix well.
Much of the power with integers apparently comes from an iterative form that I decided to call a binary quadratic Diophantine iterator, or BQD Iterator for short. And here is a key post:
BQD Iterator is a very surprising tool
The math apparently, as I still hedge, uses just a few simple tools to control all behavior of integers.
Such stunning efficiency with a few simple mathematical tools with power over all of number theory? Is the kind of logical thing which I find appealing.
James Harris
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