Conclusion: My guess was probably right
Apr. 12th, 2011 01:02 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoy(OK, this will probably be followable by all of no-one. Oh well; I can explain later...)
So I've drawn up the rewrite tables (all this stuff to be uploaded later when I am not so tired), and at last we can conclude: d(73)+5 is indeed the best candidate for a stopping point, as I hypothesized months ago. (Many thanks once again to Jānis Iraids (and of couse Karlis Podnieks as well) for the wealth of numerical data they supplied. Seriously, he computed up to a trillion!) Of course, we can't prove d(73)+5 is the stopping point with our current methods - if we could, it wouldn't be! But I'd say the evidence is pretty suggestive.
One actual mistake that turned up - what we really want, as evidence that 73(3^n+1)+6 is the stopping point, is not just finding an n such that this has complexity 19+3n (which Jānis computed holds for n∈{15,16,17,20,21}; 21 was the largest checked), but finding one which cannot be written most efficiently as a +1 or a product. Unfortunately, of those 5 cases, only n=21 actually exhibits this. (I had to check this by manually plugging in numbers - Jānis wrote his program to prefer additions over multiplications, the reverse of what we did and the reverse of what we need for this. Fortunately, 73(3^n+1)+6 factors as 2 times a prime, so this was a lot less tedious than it sounds!) In any case, finding one that exhibits it is a lot better than not finding any!
Now back to rewriting the draft...
-Harry
So I've drawn up the rewrite tables (all this stuff to be uploaded later when I am not so tired), and at last we can conclude: d(73)+5 is indeed the best candidate for a stopping point, as I hypothesized months ago. (Many thanks once again to Jānis Iraids (and of couse Karlis Podnieks as well) for the wealth of numerical data they supplied. Seriously, he computed up to a trillion!) Of course, we can't prove d(73)+5 is the stopping point with our current methods - if we could, it wouldn't be! But I'd say the evidence is pretty suggestive.
One actual mistake that turned up - what we really want, as evidence that 73(3^n+1)+6 is the stopping point, is not just finding an n such that this has complexity 19+3n (which Jānis computed holds for n∈{15,16,17,20,21}; 21 was the largest checked), but finding one which cannot be written most efficiently as a +1 or a product. Unfortunately, of those 5 cases, only n=21 actually exhibits this. (I had to check this by manually plugging in numbers - Jānis wrote his program to prefer additions over multiplications, the reverse of what we did and the reverse of what we need for this. Fortunately, 73(3^n+1)+6 factors as 2 times a prime, so this was a lot less tedious than it sounds!) In any case, finding one that exhibits it is a lot better than not finding any!
Now back to rewriting the draft...
-Harry
