Chapter 8 : Continued Fraction and Pell's Equation

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Chapter 8 : Continued Fraction and Pell's Equation

Table of solutions of Pell equation and continued fraction

Solutions of Pell equation x² - ny² = ±1,

for n less than 100, no square factors
integer part of continued fraction of sqrt(n), fraction part, length
solution of Pell equation
value of x²- ny²

n	integer part	length of fraction part	fraction part	Pell x	Pell y	x²-ny²
2	1	1	2	1	1	-1
3	1	2	1, 2	2	1	1
5	2	1	4	2	1	-1
6	2	2	2, 4	5	2	1
7	2	4	1, 1, 1, 4	8	3	1
10	3	1	6	3	1	-1
11	3	2	3, 6	10	3	1
13	3	5	1, 1, 1, 1, 6	18	5	-1
14	3	4	1, 2, 1, 6	15	4	1
15	3	2	1, 6	4	1	1
17	4	1	8	4	1	-1
19	4	6	2, 1, 3, 1, 2, 8	170	39	1
21	4	6	1, 1, 2, 1, 1, 8	55	12	1
22	4	6	1, 2, 4, 2, 1, 8	197	42	1
23	4	4	1, 3, 1, 8	24	5	1
26	5	1	10	5	1	-1
29	5	5	2, 1, 1, 2, 10	70	13	-1
30	5	2	2, 10	11	2	1
31	5	8	1, 1, 3, 5, 3, 1, 1, 10	1520	273	1
33	5	4	1, 2, 1, 10	23	4	1
34	5	4	1, 4, 1, 10	35	6	1
35	5	2	1, 10	6	1	1
37	6	1	12	6	1	-1
38	6	2	6, 12	37	6	1
39	6	2	4, 12	25	4	1
41	6	3	2, 2, 12	32	5	-1
42	6	2	2, 12	13	2	1
43	6	10	1, 1, 3, 1, 5, 1, 3, 1, 1, 12	3482	531	1
46	6	12	1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12	24335	3588	1
47	6	4	1, 5, 1, 12	48	7	1
51	7	2	7, 14	50	7	1
53	7	5	3, 1, 1, 3, 14	182	25	-1
55	7	4	2, 2, 2, 14	89	12	1
57	7	6	1, 1, 4, 1, 1, 14	151	20	1
58	7	7	1, 1, 1, 1, 1, 1, 14	99	13	-1
59	7	6	1, 2, 7, 2, 1, 14	530	69	1
61	7	11	1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14	29718	3805	-1
62	7	4	1, 6, 1, 14	63	8	1
65	8	1	16	8	1	-1
66	8	2	8, 16	65	8	1
67	8	10	5, 2, 1, 1, 7, 1, 1, 2, 5, 16	48842	5967	1
69	8	8	3, 3, 1, 4, 1, 3, 3, 16	7775	936	1
70	8	6	2, 1, 2, 1, 2, 16	251	30	1
71	8	8	2, 2, 1, 7, 1, 2, 2, 16	3480	413	1
73	8	7	1, 1, 5, 5, 1, 1, 16	1068	125	-1
74	8	5	1, 1, 1, 1, 16	43	5	-1
77	8	6	1, 3, 2, 3, 1, 16	351	40	1
78	8	4	1, 4, 1, 16	53	6	1
79	8	4	1, 7, 1, 16	80	9	1
82	9	1	18	9	1	-1
83	9	2	9, 18	82	9	1
85	9	5	4, 1, 1, 4, 18	378	41	-1
86	9	10	3, 1, 1, 1, 8, 1, 1, 1, 3, 18	10405	1122	1
87	9	2	3, 18	28	3	1
89	9	5	2, 3, 3, 2, 18	500	53	-1
91	9	8	1, 1, 5, 1, 5, 1, 1, 18	1574	165	1
93	9	10	1, 1, 1, 4, 6, 4, 1, 1, 1, 18	12151	1260	1
94	9	16	1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18	2143295	221064	1
95	9	4	1, 2, 1, 18	39	4	1
97	9	11	1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18	5604	569	-1

Known results

the pattern of fraction part of continued fraction of sqrt(n) is always palindromic as

a₁, a₂, ...... , a_i, ...... , a₂, a₁, b

last "b" is twice of integer part

right part of Pell equation's value is (-1) ^k where k is the length of fraction part

Remarkable patterns

sqrt(n²+1) = n [2n]
sqrt(n²+2) = n [n, 2n]

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E-mail : kc2h-msm@asahi-net.or.jp

Hisanori Mishima