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sniffnoySo the thing that has caused all this trouble recently integer complexity stuff has been pinning down the stopping point: What is the supremum of all r for which our method allows us to classify all numbers with defect less than r?
I've been saying for quite some time that it's 19-3log_3(73), or approximately 7.28. See, we know how to handle +1s, but +6s (and +8s and +9s and +12s) are a problem. Often we can rewrite them in another form so as not to be a problem, though - but for the infinite family of numbers 73(3^n+1)+6, we don't know how to do that. And this appears to yield defects with a supremum of 19-3log_3(73). Finitely many we can handle, sure, because any individual number can have its defect calculated by computer. But an infinte family is another matter.
The first snag in demonstrating this was that our main lemma wasn't quite good enough for this - I wanted to say, for an infinite family S, we don't have to worry about S+6 until we hit r+5, where r is the sup of the defects of S. But what the main lemma actually gives us is r+3+d(2), rather lower. Getting it up to r+5 required actually examining the structure of these infinite families some more.
But then there was the second snag: Even individual numbers cause problems! Because you see, every time I'd been talking about individual numbers, I should really have been talking about individual numbers times powers of 3. We can compute the defect of n by computer, sure, but not n3^k for all k. And to get past this I had to tweak and specialize some old lemmas to handle +6s until...
...well, until we could handle +6s. And now we can. All this time trying to show we can get up to 73(3^n+1)+6, now I've shown how to go past it. It's not a stopping point at all. The method no longer has an obvious stopping point; whatever comes up, we can (it would appear) tweak the lemmas to handle it. This is all a non-issue.
Well, I've got a whole lot of text to delete now. That'll bring the page count down some...
-Harry
I've been saying for quite some time that it's 19-3log_3(73), or approximately 7.28. See, we know how to handle +1s, but +6s (and +8s and +9s and +12s) are a problem. Often we can rewrite them in another form so as not to be a problem, though - but for the infinite family of numbers 73(3^n+1)+6, we don't know how to do that. And this appears to yield defects with a supremum of 19-3log_3(73). Finitely many we can handle, sure, because any individual number can have its defect calculated by computer. But an infinte family is another matter.
The first snag in demonstrating this was that our main lemma wasn't quite good enough for this - I wanted to say, for an infinite family S, we don't have to worry about S+6 until we hit r+5, where r is the sup of the defects of S. But what the main lemma actually gives us is r+3+d(2), rather lower. Getting it up to r+5 required actually examining the structure of these infinite families some more.
But then there was the second snag: Even individual numbers cause problems! Because you see, every time I'd been talking about individual numbers, I should really have been talking about individual numbers times powers of 3. We can compute the defect of n by computer, sure, but not n3^k for all k. And to get past this I had to tweak and specialize some old lemmas to handle +6s until...
...well, until we could handle +6s. And now we can. All this time trying to show we can get up to 73(3^n+1)+6, now I've shown how to go past it. It's not a stopping point at all. The method no longer has an obvious stopping point; whatever comes up, we can (it would appear) tweak the lemmas to handle it. This is all a non-issue.
Well, I've got a whole lot of text to delete now. That'll bring the page count down some...
-Harry
