The data on conjecture 8
Jun. 30th, 2010 10:07 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoy...you know what? I don't need super-detailed ordering info below, say, 18d(2). So let's start with what looks reasonable, and worry about the details later.
I'm going to stick with my assumption from last time, that he meant ωβ rather than βω, because the latter is kind of silly if, say, β=2. Also, I'm just going to say a,b,c instead of a',b',c', because I can redefine things if I want to. :)
So! Table time![0]
OK, so far it works. But, had I been paying attention beforehand and thinking, I would have realized, of *course* it works at these low values of β. Why? Because, if the n'th number appearing in the sequence for, say, 0 mod 3, is m, then the n'th collection of infinite families appearing in the sequence for 1 mod 3 will be m*3^k+1 and similar. And similarly for 1 mod 3 and for 2 mod 3.
So, this is good reason to suspect that it works, but it's less clear that this will work when β is no longer finite. I guess the question now is, can we formalize this in a more general setting? Have we developed enough tools for this? Maybe, but I don't know.
Status: I guess I've now officially begun thinking about this! I guess I'll hold off on 9, 10, and 11 until I've actually thought about this some.
(Actually, while making the table, I realized the reason why of course it works after 2 or 3 rows, and then got a nasty shock when I made row 5 and found that it didn't work! I was very confused about this, as I was certain I had very good reason that it should work. Fortunately, I eventually realized the problem was having swapped rows 2 and 3, because I was copying them from somewhere where I had listed the 2 mod 3 values before the 1 mod 3 values.)
-Harry
[0]Reminder: Counting the leftmost column as column 0, we have that column 4 is supposed to be 2/3 of column 3, column 5 is supposed to be 3/4 of column 1, and column 6 is supposed to be 2/3 of column 2.
I'm going to stick with my assumption from last time, that he meant ωβ rather than βω, because the latter is kind of silly if, say, β=2. Also, I'm just going to say a,b,c instead of a',b',c', because I can redefine things if I want to. :)
So! Table time![0]
β a_β b_β c_β lim(a_{ωβ+n}) lim(b_{ωβ+n}) lim(c_{ωβ+n})
| | | | | | | |
0 1 1 1 2/3 3/4 2/3
1 8/9 8/9 8/9 16/27 2/3 16/27
2 64/81 5/6 5/6 5/9 16/27 5/9
3 7/9 64/81 64/81 128/243 7/12 128/243
4 20/27 7/9 7/9 14/27 5/9 14/27
5 19/27 41/54 20/27 40/81 ??? ???
OK, so far it works. But, had I been paying attention beforehand and thinking, I would have realized, of *course* it works at these low values of β. Why? Because, if the n'th number appearing in the sequence for, say, 0 mod 3, is m, then the n'th collection of infinite families appearing in the sequence for 1 mod 3 will be m*3^k+1 and similar. And similarly for 1 mod 3 and for 2 mod 3.
So, this is good reason to suspect that it works, but it's less clear that this will work when β is no longer finite. I guess the question now is, can we formalize this in a more general setting? Have we developed enough tools for this? Maybe, but I don't know.
Status: I guess I've now officially begun thinking about this! I guess I'll hold off on 9, 10, and 11 until I've actually thought about this some.
(Actually, while making the table, I realized the reason why of course it works after 2 or 3 rows, and then got a nasty shock when I made row 5 and found that it didn't work! I was very confused about this, as I was certain I had very good reason that it should work. Fortunately, I eventually realized the problem was having swapped rows 2 and 3, because I was copying them from somewhere where I had listed the 2 mod 3 values before the 1 mod 3 values.)
-Harry
[0]Reminder: Counting the leftmost column as column 0, we have that column 4 is supposed to be 2/3 of column 3, column 5 is supposed to be 3/4 of column 1, and column 6 is supposed to be 2/3 of column 2.
