Modular simplicity is fascinating

Have a much better handle now on a lot modular where recognize much of my own success with discovery has been because of modular algebra tools. Where discussed much recently as yet again explained tautological spaces, but now with a more instructional tone. And finally emphasized how represents a new discipline where the math does algebra for you. Which is really cool.

It is worth noting how odd math history becomes then, when you realize how much EASY was waiting for anyone who could discover. Like have gone on much about:

x² - Dy² = F mod N

where N can be prime or composite, and nonzero. Where I don't think it gets interesting at all until you reach N = 3 or higher, where I stick with positives. And just with THAT expression was able to figure out how to calculate the modular inverse, so have my own modular inverse method! Which means I get to mention myself with Euler and Euclid. Where extended Euclidean algorithm is how Euclid gets in there. While Euler and Fermat are the others who actually came up with their own ways to figure out modular inverses. And Euler simply extended Fermat who had an idea that only worked with a prime modulus.

Was cool idea anyway. I discuss simply with post: More overview on modular inverse

Given other things from my research is not surprising am also rethinking some notions that more are from others, like the idea that integer factorization is a hard problem. I fear it is NOT actually, with a proper modular approach.

Like have seen naive mathematical discussion looking at, say xy = C mod N, where C is some composite, ironically in my opinion noting that x and y are easily calculated with the modular inverse.

Um, yeah but you have to not know what you're doing to focus on that expression. And I guess they did not as to folks who put such things in mathematical texts. But how would they know what they did NOT know, which we know now? I like that sentence.

Advances in human thinking fascinate me now, so have ranged more widely. Having success in mathematics gives massive benefits. And simplicity with modular means much can be easily checked by others, which I think is a great thing.

Mathematics should be about what works.

Human beings can think all kinds of things.

With mathematics is possible to gain absolute certainty with absolute proof.


James Harris