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
ninteger
part
length of
fraction part
fraction partPell xPell yx2-ny2
211211-1
3121, 2211
521421-1
6222, 4521
7241, 1, 1, 4831
1031631-1
11323, 61031
13351, 1, 1, 1, 6185-1
14341, 2, 1, 61541
15321, 6411
1741841-1
19462, 1, 3, 1, 2, 8170391
21461, 1, 2, 1, 1, 855121
22461, 2, 4, 2, 1, 8197421
23441, 3, 1, 82451
26511051-1
29552, 1, 1, 2, 107013-1
30522, 101121
31581, 1, 3, 5, 3, 1, 1, 1015202731
33541, 2, 1, 102341
34541, 4, 1, 103561
35521, 10611
37611261-1
38626, 123761
39624, 122541
41632, 2, 12325-1
42622, 121321
436101, 1, 3, 1, 5, 1, 3, 1, 1, 1234825311
466121, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 122433535881
47641, 5, 1, 124871
51727, 145071
53753, 1, 1, 3, 1418225-1
55742, 2, 2, 1489121
57761, 1, 4, 1, 1, 14151201
58771, 1, 1, 1, 1, 1, 149913-1
59761, 2, 7, 2, 1, 14530691
617111, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14297183805-1
62741, 6, 1, 146381
65811681-1
66828, 166581
678105, 2, 1, 1, 7, 1, 1, 2, 5, 164884259671
69883, 3, 1, 4, 1, 3, 3, 1677759361
70862, 1, 2, 1, 2, 16251301
71882, 2, 1, 7, 1, 2, 2, 1634804131
73871, 1, 5, 5, 1, 1, 161068125-1
74851, 1, 1, 1, 16435-1
77861, 3, 2, 3, 1, 16351401
78841, 4, 1, 165361
79841, 7, 1, 168091
82911891-1
83929, 188291
85954, 1, 1, 4, 1837841-1
869103, 1, 1, 1, 8, 1, 1, 1, 3, 181040511221
87923, 182831
89952, 3, 3, 2, 1850053-1
91981, 1, 5, 1, 5, 1, 1, 1815741651
939101, 1, 1, 4, 6, 4, 1, 1, 1, 181215112601
949161, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 1821432952210641
95941, 2, 1, 183941
979111, 5, 1, 1, 1, 1, 1, 1, 5, 1, 185604569-1

Known results

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

• a1, a2, ...... , ai, ...... , 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]

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