User:Lériendil/Square and triangle superparticulars by prime subgroup

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Some shorthand notation used here:

  • Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
  • Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2] (the kth triangle superparticular).
  • Uk = Sk/S(k+1) stands for the kth ultraparticular, which has the same subgroup as Tk except in the case where k is congruent to 4 (mod 9), in which case the subgroup of Uk lacks prime 3 from that of Tk.
  • 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.

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 (2.3 and 2.3.p)

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

2-prime subgroup (2.3)

Square-particular Subgroup Comma
Ratio Smonzo
S2 2.3 4/3 [2 -1⟩
S3 2.3 9/8 [-3 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T2 2.3 3/2 [-1 1⟩ U2 2.3 32/27 [5 -3⟩

3-prime subgroups (2.3.p)

Square-particular Subgroup Comma
Ratio Smonzo
S4 L5 16/15 [4 -1 -1⟩
S5 L5 25/24 [-3 -1 2⟩
S9 L5 81/80 [-4 4 -1⟩
S7 2.3.7 49/48 [-4 -1 2⟩
S8 2.3.7 64/63 [6 -2 -1⟩
S17 2.3.17 289/288 [-5 -2 2⟩
Triangle-particular Subgroup Comma Ultraparticular Subgroup Comma
Ratio Smonzo Ratio Smonzo
T3 L5 6/5 [1 1 -1⟩ U3 L5 135/128 [-7 3 1⟩
T4 L5 10/9 [1 -2 1⟩ U4 2.5 128/125 [7 -3⟩
T7 2.3.7 28/27 [2 -3 1⟩ U7 2.3.7 1029/1024 [-10 1 3⟩

4-prime subgroups

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

Square-particular Subgroup Comma
Ratio Smonzo
S6 = T8 L7 36/35 [2 2 -1 -1⟩
S15 L7 225/224 [-5 2 2 -1⟩
S49 L7 2401/2400 [-5 -1 -2 4⟩
S10 L5.11 100/99 [2 -2 2 -1⟩
S11 L5.11 121/120 [-3 -1 -1 2⟩
S25 L5.13 625/624 [-4 -1 4 -1⟩
S26 L5.13 676/675 [2 -3 -2 2⟩
S16 L5.17 256/255 [8 -1 -1 -1⟩
S19 L5.19 361/360 [-3 -2 -1 2⟩
S24 L5.23 576/575 [6 2 -2 -1⟩
S81 L5.41 6561/6560 [-5 8 -1 -1⟩
S80 L5.79 6400/6399 [8 -4 2 -1⟩
Triangle-particular Subgroup Comma
Ratio Smonzo
T5 L7 15/14 [-1 1 1 -1⟩
T6 L7 21/20 [-2 1 -1 1⟩
T8 = S6 L7 36/35 [2 2 -1 -1⟩
T9 L5.11 45/44 [-2 2 1 -1⟩
T10 L5.11 55/54 [-1 -3 1 1⟩
T16 L5.17 136/135 [3 -3 -1 1⟩
T25 L5.13 325/324 [-2 -4 2 1⟩
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