Have been pushing myself to focus more on folks who can check the math and may be confident is correct which does allow for a post that focuses more on fun things. While I really am glad for SO many posts that tried to explain in as much detail as possible and as simply as I could as those helped me immensely as well.
So yeah there may be folks who were aware from my original paper back to its appearance in 2004 that I'd shown that established mathematical ideas could lead to apparent contradiction. Where now I explain simply as: you can declare the ring to be algebraic integers, use statements correct in that ring and ring operations yet get to a conclusion NOT true in THAT ring. Which is wild.
But oh yeah, so my path of discovery has given LOTS to human mathematics.
So now we have an explanation for prime counts close to continuous functions. Like count of primes up to 10 is 4: 2, 3, 5 and 7. And 10/ln 10 equals approximately 4.34.
While is 25 primes to 100 and 100/ln 100 is about 21.7. While 168 primes to 1000 and 1000/ln 1000 is about 144.7 and that growing lag I can easily explain. And one of my more popular posts talks:
Differential equation and prime counting
Where that is at the end of the post. And yeah for me for over a decade as figured out my way to count primes back summer of 2002 have wondered how that could apparently sit. But also yeah, have garnered global interest as have discussed. So is more like mysteriously to me there hasn't been discussion I noticed as expected.
But also now the world has a primary way to calculate the modular inverse which to me is probably THE way to do it, where talked that quite a bit including a recent post where stepped through my derivation with more reflection.
Still my biggest result by far is finding out there were more numbers to catalog!!! Which goes back to that apparent contradiction thing. And feel like give a good overview with a recent post:
Why new numbers, really?
Where there has been LOTS of other discovery as well. Will admit that while proof is great there are times I like to look at some numbers behaving as expected, so here is a link to one of my favorite posts where I collect cases where they DO, which I did just to feel better:
Some number examples
And yeah, thanks to my need to be sure of my own research came up with a functional definition of mathematical proof. Also have used my advanced methods to figure out a simpler way to reduce two variable quadratic equations, where could also get far with a special case of a three variable quadratic, plus much more!!!
Yeah am getting tired of typing. And trying not to link to tons of things. But we're talking about over two decades of discovery. This blog covers THAT and yeah you can find all of the above, if you're interested.
So why do I think math discovery is fun?
There is a joy in the answers.
We get to know things that are answers for things some of the greatest mathematicians in human history puzzled over, as they laid the foundations upon which the answers could be found. Especially appreciate Gauss of course and Euler as they had so much to do with much, but also with modular.
Modular algebra is just such a gift to humanity. Can even do algebra for us, and better than we can on our own. Yes, discovery is fun.
Am so glad for the knowledge really.
Where also yeah, there cannot be a rush with things with such impact.
Humanity will take whatever time it needs, which includes me. As years go by, feel less shocked myself. And more and more I wonder about knowledge, and how we know--or think we know things, as a species or as individuals.
James Harris
Sunday, March 17, 2019
Sunday, March 03, 2019
Thoughts on my modular inverse derivation and possible algorithm
Thankfully went through a more stepped out derivation of my method for calculating the modular inverse with post: Meta and my modular inverse method
There I finally paid more attention to the requirement from my derivation:
nF0 = F+1 mod N
Which I noted puts pressure on the math. Which can happen if F0 shares prime factors with N, as then those have to divide off, for n to exist.
That situation actually occurs in my long example where I didn't realize why then, but simply thought it was an irritation without knowing the why of it happening. And discussed a bit in this post, where now also can talk how to avoid.
Of course F0 = r(r+2my0) mod N, so you actually have control there, as you choose m and y0, which means for an algorithm can actually pick such that F0 does NOT share prime factors by preventing r+2my0 = N mod p, for any prime factor p of N. Where of course r has to be coprime already which is a tidbit I've tended to leave out stating as that is obviously required.
Oh yeah, so the derivation is focusing on F = x2 - Dy2 mod N, as a key equation and finding I can focus on r, and get to F = r(r+2my) mod N, and then self-reference using added variables n and d to get to a solution for the modular inverse:
r-1 = (n-1)(r + 2my0) - 2md mod N
With the control equation:
2mdF0 = [F0(n-1) - 1](r+2my0) mod N
And again my baseline derivation is here.
Importantly my post on the meta aspect also covered how you get to recursion, and emphasized that with next iteration the modulus must be smaller, and can cut at least in half, guaranteed--with each recursion.
Which is really cool.
Now with it figured out, to me is kind of weird how easy it all is.
James Harris
There I finally paid more attention to the requirement from my derivation:
nF0 = F+1 mod N
Which I noted puts pressure on the math. Which can happen if F0 shares prime factors with N, as then those have to divide off, for n to exist.
That situation actually occurs in my long example where I didn't realize why then, but simply thought it was an irritation without knowing the why of it happening. And discussed a bit in this post, where now also can talk how to avoid.
Of course F0 = r(r+2my0) mod N, so you actually have control there, as you choose m and y0, which means for an algorithm can actually pick such that F0 does NOT share prime factors by preventing r+2my0 = N mod p, for any prime factor p of N. Where of course r has to be coprime already which is a tidbit I've tended to leave out stating as that is obviously required.
Oh yeah, so the derivation is focusing on F = x2 - Dy2 mod N, as a key equation and finding I can focus on r, and get to F = r(r+2my) mod N, and then self-reference using added variables n and d to get to a solution for the modular inverse:
r-1 = (n-1)(r + 2my0) - 2md mod N
With the control equation:
2mdF0 = [F0(n-1) - 1](r+2my0) mod N
And again my baseline derivation is here.
Importantly my post on the meta aspect also covered how you get to recursion, and emphasized that with next iteration the modulus must be smaller, and can cut at least in half, guaranteed--with each recursion.
Which is really cool.
Now with it figured out, to me is kind of weird how easy it all is.
James Harris
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