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.

integer	prime limit	factorization	monzo
2	2	2	[1⟩
3	3	3	[0 1⟩

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⟩
S31	L5.31	961/960	[-6 -1 -1 2⟩
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⟩

5-prime subgroups

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

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S21	L11	441/440	[-3 2 -1 2 -1⟩
S55	L11	3025/3024	[-4 -3 2 -1 2⟩
S99	L11	9801/9800	[-3 4 -2 -2 2⟩
S14	L7.13	196/195	[2 -1 -1 2 -1⟩
S64	L7.13	4096/4095	[12 -2 -1 -1 -1⟩
S35 = T49	L7.17	1225/1224	[-3 -2 2 2 -1⟩
S50	L7.17	2500/2499	[2 -1 4 -2 -1⟩
S20	L7.19	400/399	[4 -1 2 -1 -1⟩
S161	L7.23	25921/25920	[-6 -4 -1 2 2⟩
S29	L7.29	841/840	[-3 -1 -1 -1 2⟩
S125	L7.31	15625/15624	[-3 -2 6 -1 -1⟩
S36	L7.37	1296/1295	[4 4 -1 -1 -1⟩
S41	L7.41	1681/1680	[-4 -1 -1 -1 2⟩
S244	L7.61	59536/59535	[4 -5 -1 -2 2⟩
S71	L7.71	5041/5040	[-4 -2 -1 -1 2⟩
S225	L7.113	50625/50624	[-6 4 -1 4 -1⟩
S126	L7.127	15876/15875	[2 4 -3 2 -1⟩
S224	L7.223	50176/50175	[10 -2 -2 2 -1⟩
S251	L7.251	59536/59535	[-3 -2 -3 -1 2⟩
S449	L7.449	201601/201600	[-7 -2 -2 -1 2⟩
S4375	L7.547	19140625/19140624	[-4 -7 8 2 -1⟩
S2401	L7.1201	5764801/5764800	[-5 -1 -2 8 -1⟩
S2400	L7.2399	5760000/5759999	[10 2 4 -4 -1⟩
S4801	L7.4801	23049601/23049600	[-7 -1 -2 -4 2⟩

Triangle-particular	Subgroup	Comma		Ultraparticular	Subgroup	Comma
		Ratio	Smonzo			Ratio	Smonzo
T13	L7.13	91/90	[-1 -2 -1 1 1⟩	U13	2.5.7.13	10985/10976	[-5 1 -3 3⟩
T14	L7.13	105/104	[-3 1 1 1 -1⟩	U14	L7.13	43904/43875	[7 -3 -3 3 -1⟩
T26	L7.13	351/350	[-1 3 -2 -1 1⟩	U26	L7.13	492128/492075	[5 -9 -2 1 3⟩
T15	L7.17	120/119	[3 1 1 -1 -1⟩	U15	L7.17	57375/57344	[-13 3 3 -1 1⟩
T49 = S35	L7.17	1225/1224	[-3 -2 2 2 -1⟩	U49	2.5.7.17	2000033/2000000	[-7 -6 6 1⟩
T19	L7.19	190/189	[1 -3 1 -1 1⟩	U19	L7.19	48013/48000	[-7 -1 -3 1 3⟩
T28	L7.29	406/405	[1 -4 -1 1 1⟩	U28	L7.29	219520/219501	[7 -2 1 3 -3⟩
T126	L7.127	8001/8000	[-6 2 -3 1 1⟩	U126	L7.127	256048128/256047875	[10 6 -3 3 -3⟩

11-add-one-limit

L5.11.p subgroups

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S65	L13(-7)	4225/4224	[-7 -1 2 -1 2⟩
S45	L5.11.23	2025/2024	[-3 4 2 -1 -1⟩
S44	L5.11.43	1936/1935	[4 -2 -1 2 -1⟩
S54	L5.11.53	2916/2915	[2 6 -1 -1 -1⟩
S121	L5.11.61	14641/14640	[-4 -1 -1 4 -1⟩
S89	L5.11.89	7921/7920	[-4 -2 -1 -1 2⟩
S485	L5.11.97	235225/235224	[-3 -5 2 -2 2⟩
S100	L5.11.101	10000/9999	[4 -2 4 -1 -1⟩
S109	L5.11.109	11881/11880	[-3 -3 -1 -1 2⟩
S199	L5.11.199	39601/39600	[-4 -2 -2 -1 2⟩
S241	L5.11.241	58081/58080	[-5 -1 -1 -2 2⟩

Triangle-particular	Subgroup	Comma		Ultraparticular	Subgroup	Comma
		Ratio	Smonzo			Ratio	Smonzo
T11	L13(-7)	66/65	[1 1 -1 1 -1⟩	U11	L13(-7)	17303/17280	[7 3 1 -3 -1⟩
T23	L5.11.23	276/275	[2 1 -2 -1 1⟩	U23	L5.11.23	304175/304128	[-10 -3 2 -1 3⟩
T31	L5.11.31	496/495	[4 -2 -1 -1 1⟩	U31	2.5.11.31	327701/327680	[-16 -1 1 3⟩
T241	L5.11.241	29161/29160	[-4 -6 -1 2 1⟩	U241	L5.11.241	1133799201/1133799040	[-7 4 -1 -6 3⟩

No-fives (L11(-5).p) subgroups

Square-particular	Subgroup	Comma
		Ratio	Smonzo
S22	L11(-5).23	484/483	[2 -1 -1 2 -1⟩
S43	L11(-5).43	1849/1848	[-3 -1 -1 -1 2⟩
S98	L11(-5).97	9604/9603	[2 -2 4 -1 -1⟩
S197	L11(-5).197	38809/38808	[-3 -2 -2 -1 2⟩

Triangle-particular	Subgroup	Comma		Ultraparticular	Subgroup	Comma
		Ratio	Smonzo			Ratio	Smonzo
T12	L13(-5)	78/77	[1 1 -1 -1 1⟩	U12	L13(-5)	24192/24167	[7 3 1 -1 -3⟩
T22	L11(-5).23	253/252	[-2 -2 -1 1 1⟩	U22	2.7.11.23	85184/85169	[6 -1 3 -3⟩
T97	L11(-5).97	4753/4752	[-4 -3 2 -1 1⟩	U97	L11(-5).97	30118209/30118144	[-8 1 -6 1 3⟩

See also

• Third-superparticulars and semiparticulars by prime subgroup