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sniffnoyHere it is.
For now, I'm going to skip a long explanation of the contents of this paper. Perhaps I'll come back and do it later, but I'm guessing a lot of the people reading this already know a fair bit about it.
But goddamn what a relief it is to finally have this done. I mean it's not done, of course, I'm sure there will be changes necessary for publication. But it's up. It's out there.
We weren't sure whether to include section 5, and in particular the "off-by-one" theorem, but, ultimately, we included it. Otherwise it would have had to be a separate paper, which just makes for more headache, you know? (There is other stuff that got cut for length -- by which I mean, never actually written -- that will have to be a separate paper, but, that was never realistically making it in here.)
I actually haven't said as much on the internet about the contents of this paper as I have about previous paper... really, a lot of the stuff regarding integer complexity that the ideas in here led to, I just haven't really talked about so much. Well, I should remedy that. Maybe in the coming days or weeks I'll go back, and post about some of that stuff, that Arias de Reyna and I have coming down the pipeline in the wake of this. I mean, I imagine it'll be a while before those get written up properly, but I'd like to talk about it some.
But like the thing about this paper, that I've wanted to get this out there so bad is that like... I think this is kind of the best paper I've written, y'know? And it might well be the best one I write for a long time. Not in terms of the strength of the results (we've got stronger ones coming), or the quality of the exposition, but in terms of, tying things together. This paper just really ties everything together really nicely -- answering simultaneously what look like two unrelated conjectures -- and solves problems rather than raises them. It does raise new problems, of course, but ones that don't seem as essential as the ones it solves. And it finally resolves all of Arias de Reyna's old conjectures!
So, that's it. It's up. And I'm going to stop here now. Maybe more on this later.
For now, I'm going to skip a long explanation of the contents of this paper. Perhaps I'll come back and do it later, but I'm guessing a lot of the people reading this already know a fair bit about it.
But goddamn what a relief it is to finally have this done. I mean it's not done, of course, I'm sure there will be changes necessary for publication. But it's up. It's out there.
We weren't sure whether to include section 5, and in particular the "off-by-one" theorem, but, ultimately, we included it. Otherwise it would have had to be a separate paper, which just makes for more headache, you know? (There is other stuff that got cut for length -- by which I mean, never actually written -- that will have to be a separate paper, but, that was never realistically making it in here.)
I actually haven't said as much on the internet about the contents of this paper as I have about previous paper... really, a lot of the stuff regarding integer complexity that the ideas in here led to, I just haven't really talked about so much. Well, I should remedy that. Maybe in the coming days or weeks I'll go back, and post about some of that stuff, that Arias de Reyna and I have coming down the pipeline in the wake of this. I mean, I imagine it'll be a while before those get written up properly, but I'd like to talk about it some.
But like the thing about this paper, that I've wanted to get this out there so bad is that like... I think this is kind of the best paper I've written, y'know? And it might well be the best one I write for a long time. Not in terms of the strength of the results (we've got stronger ones coming), or the quality of the exposition, but in terms of, tying things together. This paper just really ties everything together really nicely -- answering simultaneously what look like two unrelated conjectures -- and solves problems rather than raises them. It does raise new problems, of course, but ones that don't seem as essential as the ones it solves. And it finally resolves all of Arias de Reyna's old conjectures!
So, that's it. It's up. And I'm going to stop here now. Maybe more on this later.
no subject
Date: 2021-11-02 11:48 pm (UTC)no subject
Date: 2021-11-03 01:31 am (UTC)I mean to be clear the paper isn't 8 years old, just the theorems. Juan Arias de Reyna and I came up with two different proofs of Conjecture 8 and Conjecture 2 way back then, but, for various reasons, we didn't get it written up until now. If you're wondering which of the two proofs we used, though -- the answer is neither! We went with a third proof I only came up with a few months ago. Our older proofs were much worse. :P
Also, in writing this up, it became clear that although I'd written down the "off-by-one" theorem in my notes to Arias de Reyna, I'd never actually fully proven it; actually proving it involved a number of details I simply hadn't considered at the time.
no subject
Date: 2021-11-03 09:31 pm (UTC)