User:Lériendil/Third-superparticulars and semiparticulars by prime subgroup: Difference between revisions

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Note that not all members of G''k'' and R''k'' are superparticular. In particular, G(3''k'') is throdd-particular, and R(4''k'') 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.
Note that not all members of G''k'' and R''k'' are superparticular. In particular, G(3''k'') is throdd-particular, and R(4''k'') 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.
This list eventually aims to be complete to the 17-add-two-limit and the 29-add-one-limit, i.e. the union of the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit, and the class of subgroups with at most two primes greater than 17, which is a superset of the 23-limit.


== 2- and 3-prime subgroups (2.p and 2.3.p) ==
== 2- and 3-prime subgroups (2.p, 2.3.p, and 2.5.p) ==
Note that the following list is ''complete'' and the insertion of higher primes will add no new inclusions to it.
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.


=== 2-prime subgroups (2.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|-
| G4 = R3
| 2.5
| [[5/4]]
| {{monzo| -2 1 }}
|-
| G5
| 2.7
| [[8/7]]
| {{monzo| 3 -1 }}
|}
=== 3-prime subgroups (2.3.p) ===
{| class="wikitable center-1 center-2 center-5 center-6"
{| class="wikitable center-1 center-2 center-5 center-6"
|-
|-
Line 27: Line 49:
! Ratio
! Ratio
! Smonzo
! Smonzo
|-
|
|
|
|
| R3 = G4
| '''2.5'''
| [[5/4]]
| {{monzo| -2 1 }}
|-
| G4 = R3
| '''2.5'''
| [[5/4]]
| {{monzo| -2 1 }}
! colspan="4" |
|-
|-
| G7 = S4
| G7 = S4
Line 52: Line 59:
| {{monzo| -4 4 -1 }}
| {{monzo| -4 4 -1 }}
|-
|-
| G5
|
| '''2.7'''
|
| [[8/7]]
|
| {{monzo| 3 -1 }}
|
| R5 = T7
| R5 = T7
| [[2.3.7 subgroup|2.3.7]]
| [[2.3.7 subgroup|2.3.7]]
Line 70: Line 77:
| {{monzo| -1 5 -2 }}
| {{monzo| -1 5 -2 }}
|}
|}
=== 3-prime subgroups (2.5.p) ===
{| class="wikitable center-1 center-2"
|-
! rowspan="2" | Superparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
|-
| R6
| 2.5.7
| [[50/49]]
| {{monzo| 1 2 -2 }}
|-
| G14
| 2.5.13
| [[65/64]]
| {{monzo| -6 1 1 }}
|}
== 4-prime subgroups with threes ==
Note that the following lists are ''complete'' and the insertion of higher primes will add no new inclusions to them.
=== 5-add-one-limit (L5.p) ===
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G8 = T6
| [[7-limit|L7]]
| [[21/20]]
| {{monzo| -2 1 -1 1 }}
|
|
|
|
|-
| G26 = S15
| [[7-limit|L7]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| R26
| [[7-limit|L7]]
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
|-
| G11
| L5.13
| [[40/39]]
| {{monzo| 3 -1 1 -1 }}
| R11 = T25
| L5.13
| [[325/324]]
| {{monzo| -2 -4 2 1 }}
|-
|
|
|
|
| R14 = S26
| L5.13
| [[676/675]]
| {{monzo| 2 -3 -2 2 }}
|-
| G17
| L5.19
| [[96/95]]
| {{monzo| 5 1 -1 -1 }}
| R17
| L5.19
| [[1216/1215]]
| {{monzo| 6 -5 -1 1 }}
|}
=== 2.3.13.p subgroups ===
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G25
| 2.3.13.23
| [[208/207]]
| {{monzo| 4 -2 1 -1 }}
| R25
| 2.3.13.23
| [[3888/3887]]
| {{monzo| 4 5 -2 -1 }}
|}
== 4-prime no-threes subgroups and 5-prime subgroups ==
In the tables that follow, no-threes subgroups will be indicated in '''bold'''.
=== 5-add-two-limit (L5.p.q) ===
==== 7-add-one-limit (L7.p) ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
|
|
|
|
| R9 = G23
| '''L11(-3)'''
| [[176/175]]
| {{monzo| 4 -2 -1 1 }}
|-
| G13
| '''L11(-3)'''
| [[56/55]]
| {{monzo| 3 -1 1 -1 }}
| R13
| [[11-limit|L11]]
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
|-
| G23 = R9
| '''L11(-3)'''
| [[176/175]]
| {{monzo| 4 -2 -1 1 }}
| R23 = S55
| [[11-limit|L11]]
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
|-
| G34
| [[11-limit|L11]]
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| R34 = S99
| [[11-limit|L11]]
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
|-
| G16
| L7.17
| [[85/84]]
| {{monzo| -2 -1 1 -1 1 }}
|
|
|
|
|-
| G19 = T15
| L7.17
| [[120/119]]
| {{monzo| 3 1 1 -1 -1 }}
| R19
| L7.17
| [[1701/1700]]
| {{monzo| -2 5 -2 1 -1 }}
|-
| G22
| '''2.5.7.23'''
| [[161/160]]
| {{monzo| -5 -1 1 1 }}
| R22
| L7.23
| [[2646/2645]]
| {{monzo| 1 3 -1 2 -2 }}
|-
| G47
| L7.23
| [[736/735]]
| {{monzo| 5 -1 -1 -2 1 }}
| R47 = S161
| L7.23
| [[25921/25920]]
| {{monzo| -6 -4 -1 2 2 }}
|-
| G29
| L7.31
| [[280/279]]
| {{monzo| 3 -2 1 1 -1 }}
| R29
| L7.31
| [[6076/6075]]
| {{monzo| 2 -2 -5 2 1 }}
|-
| G62
| L7.61
| [[1281/1280]]
| {{monzo| -8 1 -1 1 1 }}
| R62 = S244
| L7.61
| [[59536/59535]]
| {{monzo| 4 -5 -1 -2 2 }}
|-
| G82
| L7.83
| [[2241/2240]]
| {{monzo| -6 3 -1 -1 1 }}
| R82
| L7.83
| [[137781/137780]]
| {{monzo| -2 9 -1 1 -2 }}
|}
==== L5.11.p subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G31
| '''2.5.11.29'''
| [[320/319]]
| {{monzo| 6 1 -1 -1 }}
| R31
| L5.11.29
| [[7425/7424]]
| {{monzo| -8 3 2 1 -1 }}
|-
| G46
| L5.11.47
| [[705/704]]
| {{monzo| -6 1 1 -1 1 }}
| R46
| L5.11.47
| [[24300/24299]]
| {{monzo| 2 5 2 -1 -2 }}
|-
| G98
| L5.11.97
| [[3201/3200]]
| {{monzo| -7 1 -2 1 1 }}
| R98 = S485
| L5.11.97
| [[235225/235224]]
| {{monzo| -3 -5 2 -2 2 }}
|-
| G241
| L5.11.239
| [[19360/19359]]
| {{monzo| 5 -4 1 2 -1 }}
| R241
| L5.11.239
| [[3499200/3499199]]
| {{monzo| 6 7 2 -4 -1 }}
|}
==== L5.13.p subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G28
| L5.13.29
| [[261/260]]
| {{monzo| -2 2 -1 -1 1 }}
|
|
|
|
|-
| G38
| L5.13.37
| [[481/480]]
| {{monzo| -5 -1 -1 1 1 }}
| R38
| L5.13.37
| [[13690/13689]]
| {{monzo| 1 -4 1 -2 2 }}
|}
==== L5.17.p subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
|
|
|
|
| R18
| '''2.5.17.19'''
| [[1445/1444]]
| {{monzo| -2 1 2 -2 }}
|-
| G49
| '''2.5.17.47'''
| [[800/799]]
| {{monzo| 5 2 -1 -1 }}
| R49
| L5.17.47
| [[29376/29375]]
| {{monzo| 6 3 -4 1 -1 }}
|-
| G52
| L5.17.53
| [[901/900]]
| {{monzo| -2 -2 -2 1 1 }}
|
|
|
|
|}
==== Higher-prime subgroups ====
{| class="wikitable center-1 center-2 center-5 center-6"
|-
! rowspan="2" | Third-particular
! rowspan="2" | Subgroup
! colspan="2" | Comma
! rowspan="2" | Semiparticular
! rowspan="2" | Subgroup
! colspan="2" | Comma
|-
! Ratio
! Smonzo
! Ratio
! Smonzo
|-
| G74
| L5.19.73
| [[1825/1824]]
| {{monzo| -3 -2 2 -1 1 }}
| R74
| L5.19.73
| [[101251/101250]]
| {{monzo| -1 -4 -4 1 2 }}
|-
| G73
| L5.37.71
| [[1776/1775]]
| {{monzo| 4 1 -2 1 -1 }}
| R73
| L5.37.71
| [[97200/97199]]
| {{monzo| 4 5 2 -2 -1 }}
|-
| G161
| L5.53.163
| [[8640/8639]]
| {{monzo| 6 3 1 -1 -1 }}
| R161
| L5.53.163
| [[1043200/1043199]]
| {{monzo| 8 -9 2 -1 1 }}
|-
| G242
| L5.61.241
| [[19521/19520]]
| {{monzo| -6 4 -1 -1 1 }}
| R242
| L5.61.241
| [[3542941/3542940]]
| {{monzo| -2 -11 -1 1 2 }}
|}
=== No-fives subgroups ===
==== 7-add-two-limit (2.3.7.p.q) ====
==== Higher primes ====


== See also ==
== See also ==
* [[User:Lériendil/Square_and_triangle_superparticulars_by_prime_subgroup|Square and triangle superparticulars by prime subgroup]]
* [[User:Lériendil/Square_and_triangle_superparticulars_by_prime_subgroup|Square and triangle superparticulars by prime subgroup]]

Latest revision as of 04:34, 26 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 17-add-two-limit and the 29-add-one-limit, i.e. the union of the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit, and the class of subgroups with at most two primes greater than 17, which is a superset of the 23-limit.

2- and 3-prime subgroups (2.p, 2.3.p, and 2.5.p)

Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.

2-prime subgroups (2.p)

Third-particular Subgroup Comma
Ratio Smonzo
G4 = R3 2.5 5/4 [-2 1⟩
G5 2.7 8/7 [3 -1⟩

3-prime subgroups (2.3.p)

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
G7 = S4 L5 16/15 [4 -1 -1⟩ R7 = S9 L5 81/80 [-4 4 -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⟩

3-prime subgroups (2.5.p)

Superparticular Subgroup Comma
Ratio Smonzo
R6 2.5.7 50/49 [1 2 -2⟩
G14 2.5.13 65/64 [-6 1 1⟩

4-prime subgroups with threes

Note that the following lists are complete and the insertion of higher primes will add no new inclusions to them.

5-add-one-limit (L5.p)

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
G8 = T6 L7 21/20 [-2 1 -1 1⟩
G26 = S15 L7 225/224 [-5 2 2 -1⟩ R26 L7 4375/4374 [-1 -7 4 1⟩
G11 L5.13 40/39 [3 -1 1 -1⟩ R11 = T25 L5.13 325/324 [-2 -4 2 1⟩
R14 = S26 L5.13 676/675 [2 -3 -2 2⟩
G17 L5.19 96/95 [5 1 -1 -1⟩ R17 L5.19 1216/1215 [6 -5 -1 1⟩

2.3.13.p subgroups

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
G25 2.3.13.23 208/207 [4 -2 1 -1⟩ R25 2.3.13.23 3888/3887 [4 5 -2 -1⟩

4-prime no-threes subgroups and 5-prime subgroups

In the tables that follow, no-threes subgroups will be indicated in bold.

5-add-two-limit (L5.p.q)

7-add-one-limit (L7.p)

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
R9 = G23 L11(-3) 176/175 [4 -2 -1 1⟩
G13 L11(-3) 56/55 [3 -1 1 -1⟩ R13 L11 540/539 [2 3 1 -2 -1⟩
G23 = R9 L11(-3) 176/175 [4 -2 -1 1⟩ R23 = S55 L11 3025/3024 [-4 -3 2 -1 2⟩
G34 L11 385/384 [-7 -1 1 1 1⟩ R34 = S99 L11 9801/9800 [-3 4 -2 -2 2⟩
G16 L7.17 85/84 [-2 -1 1 -1 1⟩
G19 = T15 L7.17 120/119 [3 1 1 -1 -1⟩ R19 L7.17 1701/1700 [-2 5 -2 1 -1⟩
G22 2.5.7.23 161/160 [-5 -1 1 1⟩ R22 L7.23 2646/2645 [1 3 -1 2 -2⟩
G47 L7.23 736/735 [5 -1 -1 -2 1⟩ R47 = S161 L7.23 25921/25920 [-6 -4 -1 2 2⟩
G29 L7.31 280/279 [3 -2 1 1 -1⟩ R29 L7.31 6076/6075 [2 -2 -5 2 1⟩
G62 L7.61 1281/1280 [-8 1 -1 1 1⟩ R62 = S244 L7.61 59536/59535 [4 -5 -1 -2 2⟩
G82 L7.83 2241/2240 [-6 3 -1 -1 1⟩ R82 L7.83 137781/137780 [-2 9 -1 1 -2⟩

L5.11.p subgroups

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
G31 2.5.11.29 320/319 [6 1 -1 -1⟩ R31 L5.11.29 7425/7424 [-8 3 2 1 -1⟩
G46 L5.11.47 705/704 [-6 1 1 -1 1⟩ R46 L5.11.47 24300/24299 [2 5 2 -1 -2⟩
G98 L5.11.97 3201/3200 [-7 1 -2 1 1⟩ R98 = S485 L5.11.97 235225/235224 [-3 -5 2 -2 2⟩
G241 L5.11.239 19360/19359 [5 -4 1 2 -1⟩ R241 L5.11.239 3499200/3499199 [6 7 2 -4 -1⟩

L5.13.p subgroups

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
G28 L5.13.29 261/260 [-2 2 -1 -1 1⟩
G38 L5.13.37 481/480 [-5 -1 -1 1 1⟩ R38 L5.13.37 13690/13689 [1 -4 1 -2 2⟩

L5.17.p subgroups

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
R18 2.5.17.19 1445/1444 [-2 1 2 -2⟩
G49 2.5.17.47 800/799 [5 2 -1 -1⟩ R49 L5.17.47 29376/29375 [6 3 -4 1 -1⟩
G52 L5.17.53 901/900 [-2 -2 -2 1 1⟩

Higher-prime subgroups

Third-particular Subgroup Comma Semiparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
G74 L5.19.73 1825/1824 [-3 -2 2 -1 1⟩ R74 L5.19.73 101251/101250 [-1 -4 -4 1 2⟩
G73 L5.37.71 1776/1775 [4 1 -2 1 -1⟩ R73 L5.37.71 97200/97199 [4 5 2 -2 -1⟩
G161 L5.53.163 8640/8639 [6 3 1 -1 -1⟩ R161 L5.53.163 1043200/1043199 [8 -9 2 -1 1⟩
G242 L5.61.241 19521/19520 [-6 4 -1 -1 1⟩ R242 L5.61.241 3542941/3542940 [-2 -11 -1 1 2⟩

No-fives subgroups

7-add-two-limit (2.3.7.p.q)

Higher primes

See also

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