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, 2.3.p, and 2.5.p) ==
== 2- and 3-prime subgroups (2.p, 2.3.p, and 2.5.p) ==
Line 25: Line 25:
|-
|-
| G4 = R3
| G4 = R3
| '''2.5'''
| 2.5
| [[5/4]]
| [[5/4]]
| {{monzo| -2 1 }}
| {{monzo| -2 1 }}
|-
|-
| G5
| G5
| '''2.7'''
| 2.7
| [[8/7]]
| [[8/7]]
| {{monzo| 3 -1 }}
| {{monzo| 3 -1 }}
Line 89: Line 89:
|-
|-
| R6
| R6
| '''2.5.7'''
| 2.5.7
| [[50/49]]
| [[50/49]]
| {{monzo| 1 2 -2 }}
| {{monzo| 1 2 -2 }}
|-
|-
| G14
| G14
| '''2.5.13'''
| 2.5.13
| [[65/64]]
| [[65/64]]
| {{monzo| -6 1 1 }}
| {{monzo| -6 1 1 }}
Line 100: Line 100:


== 4-prime subgroups with threes ==
== 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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