With over 13 years to consider since publication of my paper is not surprising have well-worked explanations now for significance.
What I discovered is if you start by declaring the ring of algebraic integers, have a beginning statement in that ring and take valid algebraic steps from it, you can still reach a conclusion--not true in the ring of algebraic integers!!! Which I now say is like, declaring evens and then dividing 6 by 2 to get 3.
It is a coverage problem, which reveals finally after more than a century since was first believed that roots of monic polynomials with integer coefficients could encapsulate ALL integer-like number that mathematics had remained more subtle than many realized.
My original argument relied on a cubic, but simplified to a quadratic with same result. And even better, found a different path to the same conclusion with the generalized factorization in the complex plane:
P(x) = (g1(x) + 1)(g2(x) + 2)
where P(x) is a primitive quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.
Being able to go to the field of complex numbers removed needing to worry about problems with ring of algebraic integers, and definitively reveals the same conclusions as before. A key post is here which steps through the short argument.
So two paths revealing something easily explained in a paragraph, and so much drama around the result, back in 2004. From an academic perspective, with the logical conclusion that NO paper should be declared in the ring of algebraic integers, would guess is a nightmare. Brings into question any mathematical paper declared in that ring. But am NOT an academic. Am a mathematical discoverer, but not a mathematician. So have certain dispassion there.
Is not my only discovery where there has been a sad delay either. Also over a decade ago found a simple approach to counting prime numbers, talk it a lot on the blog, but discuss things fully here, which will reference for this post, giving key points. And my approach tweaks older ideas giving a vastly simpler sieve form:
P(x,n) = [x] - 1 - sum for j=1 to n of {P([x/p_j],j-1) - (j-1)}
where if n is greater than the count of primes up to and including sqrt(x) then n is reset to that count.
But more importantly, the fully mathematized form relies on a difference equation, from which a partial differential equation is readily found. But to make the summation with the difference equation give the correct prime count it has to be limited. You have to deliberately stop it from going to primes above sqrt(x), which of course goes away with the continuous function!
P'y(x,y) = -(P(x/y,y) - P(y, sqrt(y))) P'(y, sqrt(y))
And our connection between the count of prime numbers and a differential form is complete.
Since with the continuous function you're no longer stopping it deliberately, it would keep going, subtracting more, and have a REASON for lagging the count of primes.
And that is what's seen with what we can assume are approximations, with x/ln x, for instance 100/ln 100 equals approximately 21.71, which lags behind the prime count of 25. And 1000/ln 1000 is approximately 144.76 which lags behind the count of primes up to 1000, which is 168. Guessing then it would never catch it.
Does help to be curious though. Am not talking things fresh here, but things have been talking for YEARS. And I wanted to talk both cases because earlier mathematicians had something important--curiousity.
They weren't looking for some way to pursue an academic career.
There was a time when mathematicians were hungry for knowledge.
They were people who really wanted to know things like, how do you cover all integer-like numbers? Gauss had intrigued them with a+bi where 'a' and 'b' are integers. How far could you go? And they DID try. Just turns out mathematics can be wonderfully subtle. Took doing some things different to figure out what I did.
And with prime numbers, Euler, Gauss and yes, Riemann among others were curious!!!
They wondered WHY prime counts could possibly have this seeming connection with continuous functions.
And now we know. Our people in our times have known for years now.
But I guess there is no academic career potential there for others, or things would have gone differently, eh? I disdain such things. So what good is the knowledge then, to the modern mathematician?
Where did that basic curiosity go?
Could talk more things, but feel like have made my point. That I could fill this post with more major discoveries is simply further indictment of a system that lost its way. When mathematicians forgot that the point, is in discovery.
I love discovery. The people who do? Will always be the ones who move things forward, for a planet of humans, who need knowledge.
James Harris
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