End of this series: 9, 10, 11
Jul. 2nd, 2010 09:25 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyThis is where things start to look complicated. Let's see if they actually are.
Once again, wherever Arias has written βω, I'm going to assume he meant ωβ. (Do some people use the reverse convention for ordinal multiplication, or something?)
Google Translate seems to be having a little more trouble with these ones, but I think with some common sense I can figure out what's meant.
Note - here I'm just transcribing what Arias wrote (except for the ordinal multiplication switch), so that means that for now we're back to his a,b,c, not the ones I used last time.
Well, no, one more thing stands out as "obviously a mistake", so I'll fix that too - the last line of these conjectures, he talks about aγ and aβ in all three. I'm going to assume that was supposed to vary appropriately. I'm also going to multiply by 1/3 in the last case, as it doesn't seem to make sense otherwise.
Conjecture 9 (fixed): The numbers of the sequence bωβ+n that converge to aβ=b/3^a (with ||b||=3a) are the numbers of the sequences p(q3^j+1)/3^(a+j), where b=pq and ||p(q3^j+1)||=3a+3j+1, and those terms of the sporadic succession 2^(3j+2)/3^(2j+1) which are contained between supγ<βaγ and aβ.
Conjecture 10 (fixed): The numbers of the sequence cωβ+n that converge to bβ=b/3^a (with ||b||=3a+1) are the numbers of the sequences p(q3^j+1)/3^(a+j), where b=pq and ||p(q3^j+1)||=3a+3j+2, and those terms of the sporadic succession 2^(3j+1)/3^(2j) which are contained between supγ<βbγ and bβ.
Conjecture 11 (fixed): The numbers of the sequence aωβ+n that converge to cβ/3=b/3^a (with ||b||=3a-1) are the numbers of the sequences p(q3^j+1)/3^(a+j), where b=pq and ||p(q3^j+1)||=3a+3j, and those terms of the sporadic succession 2^(3j)/3^(2j) which are contained between (1/3)supγ<βcγ and cβ/3.
...hm. So part of this is just a formalization of what I was saying yesterday, and part of it appears to be, well, wrong. I could reformulate these in terms I would prefer, but I'm not going to, because I get what these are saying so I don't think that's really necessary.
For the rest of this, I'm just going to consider #9, as the rest should be analogous, but I don't want to think about how to modify appropriately.
Let's start by summing up: He's saying that the convergents to b/g(||b||)*appropriatefraction discussed previously discussed are some sporadic stuff, plus precisely what you get from the infinite families p(q*3^j+1), where pq=b, and (additional requirement). Now, we're going to immediately need an additional requirement, that ||b||=||p||+||q||, i.e., that this representation is minimal; otherwise, stuff won't line up. But I think his next requirement might cover this - he requires that ||p(q3^j+1)||=3j+||b||+1. Which looks stronger offhand; is it? A quick check says "I can't tell". Well, whatever. So far this is basically a formalization of the correspondence I mentioned before.
Now before we continue it's worth noting that while I do expect this correspondence to hold up down in the regime of +1s, I see little reason to expect it to continue to hold once +6s and such become relevant. Maybe conjecture 8 is still true, but it's pretty doubtful. I was talking to Josh yesterday and the conclusion seemed to be that this would be a good problem to not work on. If it's true, we don't have the tools to prove it; and if it's false, a disproof would require absurd amounts of computation (barring the automation that I still haven't talked about here), if we even had the tools at all.
Anyway, back to 9. 9 says the relevant numbers are those infinite families, plus some sporadic stuff. What sporadic stuff? Powers of 2. Well... that part just isn't right, except at low defects. *searches through files to find a counterexample* ...OK, let's take 32, with its corresponding sequences 32*3^k+1, etc. ||32||=10, add 1, that's 2 mod 3. Now, the range condition on those sporadic things - if you think about it, and apply conjecture 8, it's just saying, "stuff inbetween this collection of infinite families and the previous one of the same complexity mod 3". In this case, the latter would be 4*3^k+1, etc. So let's look between those two at stuff with complexity congruent to 1 mod 3, and check whether it all either is a power of 2, or actually falls into one of 32*3^k+1, etc.
First up is 320, but that's 32(3*3+1). Next is 35, which is... which is... none of those. It's not even, and it's certainly not 32*3^k+1, but there it is anyway. So that part of conjectures 9, 10, 11, is false. (Technically, this is only a counterexample to one of them; but IINM, 35, 70, and 133 together make counterexamples to all three. And there should be plenty more.)
As for the rest? Well, that's what I described above... although, seeing as the conjectures as written are trying to write down *all* the numbers in that range, removing the second part, well, offhand I'm not sure how you'd formalize the result. I guess just say that those numbers approach it, and everything else is bounded away from it?
So, conclusion:
1,3,4,5,6,7 are resolved (either proved or salvaged).
2 is unresolved.
The parts of 9,10,11 dealing with what the sporadic stuff is, is false. The rest can be folded into 8.
8 is unresolved, but too hard to handle without additional tools.
So the point of all this, is that just leaves #2 as something possibly to work on. Now enough of this.
-Harry
Once again, wherever Arias has written βω, I'm going to assume he meant ωβ. (Do some people use the reverse convention for ordinal multiplication, or something?)
Google Translate seems to be having a little more trouble with these ones, but I think with some common sense I can figure out what's meant.
Note - here I'm just transcribing what Arias wrote (except for the ordinal multiplication switch), so that means that for now we're back to his a,b,c, not the ones I used last time.
Well, no, one more thing stands out as "obviously a mistake", so I'll fix that too - the last line of these conjectures, he talks about aγ and aβ in all three. I'm going to assume that was supposed to vary appropriately. I'm also going to multiply by 1/3 in the last case, as it doesn't seem to make sense otherwise.
Conjecture 9 (fixed): The numbers of the sequence bωβ+n that converge to aβ=b/3^a (with ||b||=3a) are the numbers of the sequences p(q3^j+1)/3^(a+j), where b=pq and ||p(q3^j+1)||=3a+3j+1, and those terms of the sporadic succession 2^(3j+2)/3^(2j+1) which are contained between supγ<βaγ and aβ.
Conjecture 10 (fixed): The numbers of the sequence cωβ+n that converge to bβ=b/3^a (with ||b||=3a+1) are the numbers of the sequences p(q3^j+1)/3^(a+j), where b=pq and ||p(q3^j+1)||=3a+3j+2, and those terms of the sporadic succession 2^(3j+1)/3^(2j) which are contained between supγ<βbγ and bβ.
Conjecture 11 (fixed): The numbers of the sequence aωβ+n that converge to cβ/3=b/3^a (with ||b||=3a-1) are the numbers of the sequences p(q3^j+1)/3^(a+j), where b=pq and ||p(q3^j+1)||=3a+3j, and those terms of the sporadic succession 2^(3j)/3^(2j) which are contained between (1/3)supγ<βcγ and cβ/3.
...hm. So part of this is just a formalization of what I was saying yesterday, and part of it appears to be, well, wrong. I could reformulate these in terms I would prefer, but I'm not going to, because I get what these are saying so I don't think that's really necessary.
For the rest of this, I'm just going to consider #9, as the rest should be analogous, but I don't want to think about how to modify appropriately.
Let's start by summing up: He's saying that the convergents to b/g(||b||)*appropriatefraction discussed previously discussed are some sporadic stuff, plus precisely what you get from the infinite families p(q*3^j+1), where pq=b, and (additional requirement). Now, we're going to immediately need an additional requirement, that ||b||=||p||+||q||, i.e., that this representation is minimal; otherwise, stuff won't line up. But I think his next requirement might cover this - he requires that ||p(q3^j+1)||=3j+||b||+1. Which looks stronger offhand; is it? A quick check says "I can't tell". Well, whatever. So far this is basically a formalization of the correspondence I mentioned before.
Now before we continue it's worth noting that while I do expect this correspondence to hold up down in the regime of +1s, I see little reason to expect it to continue to hold once +6s and such become relevant. Maybe conjecture 8 is still true, but it's pretty doubtful. I was talking to Josh yesterday and the conclusion seemed to be that this would be a good problem to not work on. If it's true, we don't have the tools to prove it; and if it's false, a disproof would require absurd amounts of computation (barring the automation that I still haven't talked about here), if we even had the tools at all.
Anyway, back to 9. 9 says the relevant numbers are those infinite families, plus some sporadic stuff. What sporadic stuff? Powers of 2. Well... that part just isn't right, except at low defects. *searches through files to find a counterexample* ...OK, let's take 32, with its corresponding sequences 32*3^k+1, etc. ||32||=10, add 1, that's 2 mod 3. Now, the range condition on those sporadic things - if you think about it, and apply conjecture 8, it's just saying, "stuff inbetween this collection of infinite families and the previous one of the same complexity mod 3". In this case, the latter would be 4*3^k+1, etc. So let's look between those two at stuff with complexity congruent to 1 mod 3, and check whether it all either is a power of 2, or actually falls into one of 32*3^k+1, etc.
First up is 320, but that's 32(3*3+1). Next is 35, which is... which is... none of those. It's not even, and it's certainly not 32*3^k+1, but there it is anyway. So that part of conjectures 9, 10, 11, is false. (Technically, this is only a counterexample to one of them; but IINM, 35, 70, and 133 together make counterexamples to all three. And there should be plenty more.)
As for the rest? Well, that's what I described above... although, seeing as the conjectures as written are trying to write down *all* the numbers in that range, removing the second part, well, offhand I'm not sure how you'd formalize the result. I guess just say that those numbers approach it, and everything else is bounded away from it?
So, conclusion:
1,3,4,5,6,7 are resolved (either proved or salvaged).
2 is unresolved.
The parts of 9,10,11 dealing with what the sporadic stuff is, is false. The rest can be folded into 8.
8 is unresolved, but too hard to handle without additional tools.
So the point of all this, is that just leaves #2 as something possibly to work on. Now enough of this.
-Harry
