User:Lériendil/Third-superparticulars and semiparticulars by prime subgroup: Difference between revisions
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Created page with "Some shorthand notation used here: * S''k'' stands for ''k''^2/[(''k''-1)(''k''+1)] by standard convention (the ''k''th square superparticular). * G''k'' stands for S(''k''-1)..." |
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* G''k'' stands for S(''k''-1)*S''k''*S(''k''+1) (the ''k''th third-particular). | * G''k'' stands for S(''k''-1)*S''k''*S(''k''+1) (the ''k''th third-particular). | ||
* R''k'' stands for S(''k''-1)/S(''k''+1) (the ''k''th semiparticular). | * R''k'' stands for S(''k''-1)/S(''k''+1) (the ''k''th semiparticular). | ||
* T''k'' = S''k'' * S(''k''+1) stands for [''k''(''k''+1)/2]/[(''k''-1)(''k''+2)/2] (the ''k''th triangle superparticular). | |||
* L''p'' refers to the ''p''-limit, i.e. the subgroup of primes less than or equal to ''p''. | * L''p'' refers to the ''p''-limit, i.e. the subgroup of primes less than or equal to ''p''. | ||
* L''p''(-''q'') refers to the ''p'' limit with the prime ''q'' omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23. | * L''p''(-''q'') refers to the ''p'' limit with the prime ''q'' omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23. | ||
Revision as of 23:16, 25 July 2024
Some shorthand notation used here:
- Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
- Gk stands for S(k-1)*Sk*S(k+1) (the kth third-particular).
- Rk stands for S(k-1)/S(k+1) (the kth semiparticular).
- Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2] (the kth triangle superparticular).
- Lp refers to the p-limit, i.e. the subgroup of primes less than or equal to p.
- Lp(-q) refers to the p limit with the prime q omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.
Note that not all members of Gk and Rk are superparticular. In particular, G(3k) is throdd-particular, and R(4k) is odd-particular. Such ratios will be excluded from consideration in this chart, though they will appear on companion no-twos and no-threes pages.
This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.
2- and 3-prime subgroups (2.p and 2.3.p)
| Third-particular | Subgroup | Comma | Semiparticular | Subgroup | Comma | ||
|---|---|---|---|---|---|---|---|
| Ratio | Smonzo | Ratio | Smonzo | ||||
| R3 = G4 | 2.5 | 5/4 | [-2 1⟩ | ||||
| G4 = R3 | 2.5 | 5/4 | [-2 1⟩ | ||||
| G7 = S4 | L5 | 16/15 | [4 -1 -1⟩ | R7 = S9 | L5 | 81/80 | [-4 4 -1⟩ |
| G5 | 2.7 | 8/7 | [3 -1⟩ | R5 = T7 | 2.3.7 | 28/27 | [2 -3 1⟩ |
| G10 | 2.3.11 | 33/32 | [-5 1 1⟩ | R10 | 2.3.11 | 243/242 | [-1 5 -2⟩ |