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

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⟩

See also

• Square and triangle superparticulars by prime subgroup