On a Diophantine Equation That Generates All Integral Apollonian Gaskets

Table 1

Bend quintets for Apollonian integer disk packings for principal curvatures 1 through 15, including four indices and shifts.

		Symmetry	Curvatures					Four indices
		type	−𝐵0	𝐵1	𝐵2	𝐵3	𝐵4	𝐵	𝑘	𝑛	𝑚𝑢	Shift
1	1	Odd	1	2	2	3	3	1	1	1	0	0
2	1	Odd	2	3	6	7	7	2	1	4	0	0
3	1	Odd	3	4	12	13	13	3	1	9	0	0
4	2	Even	3	5	8	8	12	3	2	5	1	1
5	1	Odd	4	5	20	21	21	4	1	16	0	0
6	2	Even	4	8	9	9	17	4	4	5	2	1
7	1	Odd	5	6	30	31	31	5	1	25	0	0
8	2	Even	5	7	18	18	22	5	2	13	1	1
9	1	Odd	6	7	42	43	43	6	1	36	0	0
10	2	Odd	6	10	15	19	19	6	4	9	0	0
11	3	Skew	6	11	14	15	23	6	5	8	2	4/5
12	1	Odd	7	8	56	57	57	7	1	49	0	0
13	2	Even	7	9	32	32	36	7	2	25	1	1
14	3	Skew	7	12	17	20	24	7	5	10	1	2/5
15	1	Odd	8	9	72	73	73	8	1	64	0	0
16	2	Skew	8	13	21	24	28	8	5	13	1	2/5
17	3	Even	8	12	25	25	33	8	4	17	2	1
18	1	Odd	9	10	90	91	91	9	1	81	0	0
19	2	Even	9	11	50	50	54	9	2	41	1	1
20	3	Skew	9	14	26	27	35	9	5	17	2	4/5
21	4	Skew	9	18	19	22	34	9	9	10	3	2/3
22	1	Odd	10	11	110	111	111	10	1	100	0	0
23	2	Odd	10	14	35	39	39	10	4	25	0	0
24	3	Skew	10	18	23	27	35	10	8	13	2	1/2
⋯
34	6	Even*	12	25	25	28	48	12	13	13	5	10/13