What I think is my most controversial result at this time apparently pulls in others things which I think can be broadly considered to be a social problem, which is the label for this post and others like it. That social problem includes issues around my find of a coverage problem where the ring of algebraic integers can be shown to leave out certain integer-like numbers.

Explaining the coverage problem can be done with a simple factorization:

P(x) = (g

_{1}(x) + 1)(g

_{2}(x) + 2)

where P(x) is a quadratic with integer coefficients, g

_{1}(0) = g

_{2}(0) = 0, but g

_{1}(x) does not equal 0 for all x.

At the trivial level, it's very easy to get started, as of course:

P(x) = (x+1)(x+2) fits the requirements.

But if you look at non-polynomial g's then it's easy to find examples where algebraic integers are excluded, where you do not need a field.

The result here backs up a previously published result, where I should note an attempt by editors to remove the paper after publication, doing so from an electronic journal, while the original remains available where I've discussed the situation many times, where here I think is a decent reference post.

That paper served two purposes, both to highlight the coverage problem, and to give me the distinction in human history of being I'm sure the only person to be published with a formally peer reviewed and correct result, under established mathematical views, which nonetheless leads to a contradiction.

I doubt any other human will get that chance, so seized it.

I guess in some ways it's a dubious accomplishment, but I think I had a duty to grab it anyway, as no other human being will probably even have the opportunity.

Supposedly that was impossible, but this weird coverage thing made it possible. So yes, the paper IS correct by accepted standards of rigor, and it does lead to an apparent mathematical contradiction. Resolving that apparent contradiction is easy though. So it actually invalidates "established mathematical views". Human beings got some things wrong, but the math is still perfect.

In general with all of my mathematical results have taken the long view, where early on with this result and others I leaned heavily on simply reporting to mathematicians, making sure it was in their field of interest as best I could, and leaving it up to them to do the right thing. Which included journal submissions.

However, historically except for one case already mentioned, all submissions to journals have been rejected by their editors. Which was fascinating, curious, and intriguing as to the various reasons given. It is of little interest to expand upon further here.

At some point if I chose I could again send papers to journals. It just doesn't seem necessary as I can make the information publicly available myself, like on this blog.

Given that I do have a formally peer reviewed result, with the one published paper, even if the editors tried to withdraw, am officially a mathematical author, and also have the right to present as an established result. There has been no formal refutation of the paper which fulfills the requirements of such, which are that same be itself formally peer reviewed and published. There were attempts in the past at attacking these ideas, which are not worth elevating in this discussion as did not fulfill minimum requirements for mention at my level.

Any attempts, however, to refute these ideas are of continuing interest to me.

Members of the mathematical community can then by the rules consider them valid. And I can present them as valid, while I also like to caution that to my knowledge the results are not accepted by mainstream mathematicians, which is actually irrelevant.

But in terms of the social problem I find the situation intriguing, and have noted I have little motivation to end the situation.

Members of the mathematical community are presumed to have very high standards, rely on mathematical proof, and act on the basis of their own conscience and judgment.

Members of the mathematical community will of course be held by the absolute standard of the discipline that correctness rules, which is true through its long history, with no excuses.

This blog will continue as a reference for much of my mathematical research. And I like it more as a reference rather than dealing with these type issues which I see as more social and political. However, I continue to reserve the right to discuss what I see fit here, as it is my blog.

Other research results are far less controversial than the one dealing with the coverage problem, but much of the above still applies. Members of the mathematical community who may wish more from me are simply not going to get it at this time.

My assessment of the global mathematical community is that it is extremely healthy at this time, with working mathematics helping to drive a tremendous amount of global progress. I expect that will continue with little reason to believe anything of importance will interfere.

James Harris