Back to work: 5, 6, and 7
Jun. 30th, 2010 05:34 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoyWhy am I doing this, by the way? Because I found myself unable to sit down and think about any of these conjectures, because, well, they were in Spanish, and I wasn't sure what we had already done on them! So, I'm writing that up so I can actually get to work on them.
Again, I'm just going to apply Google Translate, because these don't appear to be that complicated.
Thought on notation: I started using the notation "L(n)" for minm≥n||m|| because, well, it's logarithmic. (Earlier I was using ugly stuff like f-hat or f-twiddle.) Perhaps a better name for what I currently call "g" would be "E", for exponential? (Recall: By g(k) I mean the largest number writable with k ones.) Well, I've been using "g" so far in this series, so I'll continue with that.
Also note: The rest of Arias's conjectures all deal with his sequences a_α, b_α, c_α[0]. But these sequences as he defined them don't actually exist! Fortunately, it seems pretty clear that even though we've redefined these sequences, these are the "same" sequences he was thinking of. Well, mostly, anyway - you'll see what I mean in a moment. So, all his further conjectures about these sequences will be taken as being about our equivalents...
5, 6, and 7, are really just one conjecture, split in three. First, he makes a definition: Let A be the set of all natural numbers n such that ||3jn||=3j+||n|| for all j.
Then, he makes conjectures 5, 6, and 7: The sequence a_α consists of the elements of the set {n/3||n||/3 : n∈A, ||n||≡0 (mod 3)} sorted in decreasing order. The sequence b_α consists of the elements of the set {n/3(||n||-1)/3 : n∈A, ||n||≡0 (mod 3)} sorted in decreasing order. The sequence c_α consists of the elements of the set {n/3(||n||-2)/3 : n∈A, ||n||≡0 (mod 3)} sorted in decreasing order.
Status: Well, um... gee. We *redefined* these three sequences to be those, except letting n range over all of N, instead of just in A. So, by our definition, this is false as written but easily salvaged (just change A to N). But, perhaps you could use this to argue that we *aren't* thinking of exactly the same sequences as Arias after all, and that we *should* have restricted to A, making it *true*.
Alternatively, you could note that this is another instance of Arias noting that ||3n||=||n||+3 isn't always true, but failing to realize the implications, and therefore we can't really meaningfully speculate on which definition he "would have meant" had he realized this, and it's best to just note the truth and not worry about exactly what the mistake might have been. After all, if the sequences existed as he defined them, then they *would* meet our definition. They'd meet this definition too, of course, because then we'd be in a world where ||3n||=||n||+3, and 1=0 etc, but they would quite straightforwardly meet our definition.
So, uh... yeah. I don't intend to change our definitions. Excluding numbers not in A would mean excluding possible things you can do with certain numbers of threes, which doesn't seem right, to my mind.
Well, that was pretty trivial (given what we know so far). So far I've gone through 1-7, and only #2 is something we haven't solved. I'll come back later and write up 8; I'm pretty sure it, and the remaining three after it, are things we haven't considered.
-Harry
[0]Question: Does anyone know a nice, short, word for a transfinite sequence? Whether indexed by all ordinals or just some set of ordinals? I've never heard one and I get the idea there isn't one. Because "sequence" typically suggests an N-indexed sequence. Well, in any case, the context makes it clear.
Again, I'm just going to apply Google Translate, because these don't appear to be that complicated.
Thought on notation: I started using the notation "L(n)" for minm≥n||m|| because, well, it's logarithmic. (Earlier I was using ugly stuff like f-hat or f-twiddle.) Perhaps a better name for what I currently call "g" would be "E", for exponential? (Recall: By g(k) I mean the largest number writable with k ones.) Well, I've been using "g" so far in this series, so I'll continue with that.
Also note: The rest of Arias's conjectures all deal with his sequences a_α, b_α, c_α[0]. But these sequences as he defined them don't actually exist! Fortunately, it seems pretty clear that even though we've redefined these sequences, these are the "same" sequences he was thinking of. Well, mostly, anyway - you'll see what I mean in a moment. So, all his further conjectures about these sequences will be taken as being about our equivalents...
5, 6, and 7, are really just one conjecture, split in three. First, he makes a definition: Let A be the set of all natural numbers n such that ||3jn||=3j+||n|| for all j.
Then, he makes conjectures 5, 6, and 7: The sequence a_α consists of the elements of the set {n/3||n||/3 : n∈A, ||n||≡0 (mod 3)} sorted in decreasing order. The sequence b_α consists of the elements of the set {n/3(||n||-1)/3 : n∈A, ||n||≡0 (mod 3)} sorted in decreasing order. The sequence c_α consists of the elements of the set {n/3(||n||-2)/3 : n∈A, ||n||≡0 (mod 3)} sorted in decreasing order.
Status: Well, um... gee. We *redefined* these three sequences to be those, except letting n range over all of N, instead of just in A. So, by our definition, this is false as written but easily salvaged (just change A to N). But, perhaps you could use this to argue that we *aren't* thinking of exactly the same sequences as Arias after all, and that we *should* have restricted to A, making it *true*.
Alternatively, you could note that this is another instance of Arias noting that ||3n||=||n||+3 isn't always true, but failing to realize the implications, and therefore we can't really meaningfully speculate on which definition he "would have meant" had he realized this, and it's best to just note the truth and not worry about exactly what the mistake might have been. After all, if the sequences existed as he defined them, then they *would* meet our definition. They'd meet this definition too, of course, because then we'd be in a world where ||3n||=||n||+3, and 1=0 etc, but they would quite straightforwardly meet our definition.
So, uh... yeah. I don't intend to change our definitions. Excluding numbers not in A would mean excluding possible things you can do with certain numbers of threes, which doesn't seem right, to my mind.
Well, that was pretty trivial (given what we know so far). So far I've gone through 1-7, and only #2 is something we haven't solved. I'll come back later and write up 8; I'm pretty sure it, and the remaining three after it, are things we haven't considered.
-Harry
[0]Question: Does anyone know a nice, short, word for a transfinite sequence? Whether indexed by all ordinals or just some set of ordinals? I've never heard one and I get the idea there isn't one. Because "sequence" typically suggests an N-indexed sequence. Well, in any case, the context makes it clear.
