My most challenging result thought would have help from mathematicians, which is key, as am NOT a mathematician as I routinely emphasize. And publication seemed like a good start, back about 14 years ago when a key paper highlighting the ability to appear to prove something not true with valid methods, disprovable by established number theory finally went live. The problem I'd found there was the declaration--in the ring of algebraic integers.
And talked much in detail in 2017 with post: Publication does matter
Paper is simply correct declared in the complex field without even apparent error. And importantly is also correct declared in my ring of objects.
Unfortunately the rest of the story made things more challenging in some ways, while being a boon for me from a singular research perspective. As have discussed in detail and chase the link to get more links to full story, the chief editor tried to withdraw my result, by simply deleting out of electronic file of an electronic journal. EMIS maintains the original, and I moved on.
This post will cover my primary basic research in that area primarily, as without help from the mathematical community had the vastly more difficult task of personal verification.
One area where focused was: how could I know for sure on my own? So I found a functional definition of mathematical proof and promptly put to use.
Importantly simplified also from cubics to quadratics. And first public post on this blog I note for here where that has been shown to have happened is with my wrapper theorem post in August 2007.
Wasn't until September 2008 that I further tested my use of what I call tautological spaces, which are complex identities, with my Quadratic Diophantine Theorem post, where used conditional:
c1x2 + c2xy + c3y2 = c4z2 + c5zx + c6zy
where the c's are constants, and importantly, realized could set z=1, and got my method for reducing binary quadratic Diophantine equations. So yeah, there is a research path with three variables which have not pursued.
It was by 2010 that I had the idea of focusing on a simple quadratic factorization:
P(x) = (g1(x) + 1)(g2(x) + 2)
Is interesting then had about six years to figure out how should be constructed, where yeah that 1 and 2 are benefits of research, though also kind of obvious to pick?
But looks like don't have a public post discussing until February 2015, with post talking blocking with algebraic integers where had to correct recently. However I also did my most exhaustive stepping through of original approach in the paper now using quadratics in September 2011, with post: Under the hood
And am skipping other research areas related as trying to be more focused for this post. And is interesting that gap I think from having a second path to proving the coverage problem, to what found public now on this blog.
But even then didn't really push it much until last year.
So have roughly a span of research in this area of about 14 years from point of publication which is not extraordinary I don't think. And was allowed to figure things out at a rather leisurely pace, while was also doing other research.
But that was just my own validation. I wanted to be thorough.
From perspective of a solitary researcher really cannot think of it flowing any better. Though it is intriguing was given the opportunity, which is outside scope of this post.
James Harris
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