User:Lériendil/Square and triangle superparticulars by prime subgroup

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

Note that the lists of triangle-particulars are complete and the insertion of higher primes will add no new inclusions to them. The lists of square particulars other than the "Higher primes" table are likewise complete.

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		Ultraparticular	Subgroup	Comma
		Ratio	Smonzo			Ratio	Smonzo
T5	L7	15/14	[-1 1 1 -1⟩	U5	L7	875/864	[-5 -3 3 1⟩
T6	L7	21/20	[-2 1 -1 1⟩	U6	L7	1728/1715	[6 3 -1 -3⟩
T8 = S6	L7	36/35	[2 2 -1 -1⟩	U8	L7	5120/5103	[10 -6 1 -1⟩
T9	L5.11	45/44	[-2 2 1 -1⟩	U9	L5.11	8019/8000	[-6 6 -3 1⟩
T10	L5.11	55/54	[-1 -3 1 1⟩	U10	L5.11	4000/3993	[5 -1 3 -3⟩
T25	L5.13	325/324	[-2 -4 2 1⟩	U25	L5.13	140625/140608	[-6 2 6 -3⟩
T16	L5.17	136/135	[3 -3 -1 1⟩	U16	L5.17	24576/24565	[13 1 -1 -3⟩

2.3.7.p subgroups

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S13	2.3.7.13	169/168	[-3 -1 -1 2⟩
S27	2.3.7.13	729/728	[-3 6 -1 -1⟩
S28	2.3.7.29	784/783	[4 -3 2 -1⟩
S63	2.3.7.31	3969/3968	[-7 4 2 -1⟩
S48	2.3.7.47	2304/2303	[8 2 -2 -1⟩
S97	2.3.7.97	9409/9408	[-6 -1 -2 2⟩
S127	2.3.7.127	16129/16128	[-8 -2 -1 2⟩

2.3.11.p subgroups

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S12	2.3.11.13	144/143	[4 2 -1 -1⟩
S33	2.3.11.17	1089/1088	[-6 2 -1 2⟩
S23	2.3.11.23	529/528	[-4 -1 -1 2⟩
S32	2.3.11.31	1024/1023	[10 -1 -1 -1⟩
S243	2.3.11.61	59049/59048	[-3 10 -2 -1⟩
S242	2.3.11.241	58564/58563	[2 -5 4 -1⟩

Higher primes

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S53	2.3.13.53	2809/2808	[-3 -3 -1 2⟩
S18	2.3.17.19	324/323	[2 4 -1 -1⟩
S577	2.3.17.577	332929/332928	[-7 -2 -2 2⟩
S37	2.3.19.37	1369/1368	[-3 -2 -1 2⟩
S47	2.3.23.47	2209/2208	[-5 -1 -1 2⟩

Triangle-particular	Subgroup	Comma		Ultraparticular	Subgroup	Comma
		Ratio	Smonzo			Ratio	Smonzo
T17	2.3.17.19	153/152	[-3 2 1 -1⟩	U17	2.3.17.19	93347/93312	[-7 -6 3 1⟩