table of continued fractions of $\sqrt{n}$ for

The simple continued fractions for the square roots of positive integers (which aren’t perfect powers) are non-terminating but they are periodic. In the following table, the square roots of the integers from 2 to 101 (excluding perfect powers) are listed in compact form: first the integer part followed by semicolon, then the periodic part stated once, its individual terms separated by commas. For example, the notation “14; 14, 28” for 198 means

$$\sqrt{198}=14+\frac{1}{14+\frac{1}{28+\frac{1}{14+\frac{1}{28+\mathrm{\dots }}}}},$$

where the dots mean a periodic repetition of 14 and 28 in the denominators.

$n$	Continued fraction of $\sqrt{n}$
2	1; 2
3	1; 1, 2
5	2; 4
6	2; 2, 4
7	2; 1, 1, 1, 4
8	2; 1, 4
10	3; 6
11	3; 3, 6
12	3; 2, 6
13	3; 1, 1, 1, 1, 6
14	3; 1, 2, 1, 6
15	3; 1, 6
17	4; 8
18	4; 4, 8
19	4; 2, 1, 3, 1, 2, 8
20	4; 2, 8
21	4; 1, 1, 2, 1, 1, 8
22	4; 1, 2, 4, 2, 1, 8
23	4; 1, 3, 1, 8
24	4; 1, 8
26	5; 10
27	5; 5, 10
28	5; 3, 2, 3, 10
29	5; 2, 1, 1, 2, 10
30	5; 2, 10
31	5; 1, 1, 3, 5, 3, 1, 1, 10
32	5; 1, 1, 1, 10
33	5; 1, 2, 1, 10
34	5; 1, 4, 1, 10
35	5; 1, 10
37	6; 12
38	6; 6, 12
39	6; 4, 12
40	6; 3, 12
41	6; 2, 2, 12
42	6; 2, 12
43	6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12
44	6; 1, 1, 1, 2, 1, 1, 1, 12
45	6; 1, 2, 2, 2, 1, 12
46	6; 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12
47	6; 1, 5, 1, 12
48	6; 1, 12
50	7; 14
51	7; 7, 14
52	7; 4, 1, 2, 1, 4, 14
53	7; 3, 1, 1, 3, 14
54	7; 2, 1, 6, 1, 2, 14
55	7; 2, 2, 2, 14
56	7; 2, 14
57	7; 1, 1, 4, 1, 1, 14
58	7; 1, 1, 1, 1, 1, 1, 14
59	7; 1, 2, 7, 2, 1, 14
60	7; 1, 2, 1, 14
61	7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14
62	7; 1, 6, 1, 14
63	7; 1, 14
65	8; 16
66	8; 8, 16
67	8; 5, 2, 1, 1, 7, 1, 1, 2, 5, 16
68	8; 4, 16
69	8; 3, 3, 1, 4, 1, 3, 3, 16
70	8; 2, 1, 2, 1, 2, 16
71	8; 2, 2, 1, 7, 1, 2, 2, 16
72	8; 2, 16
73	8; 1, 1, 5, 5, 1, 1, 16
74	8; 1, 1, 1, 1, 16
75	8; 1, 1, 1, 16
76	8; 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16
77	8; 1, 3, 2, 3, 1, 16
78	8; 1, 4, 1, 16
79	8; 1, 7, 1, 16
80	8; 1, 16
82	9; 18
83	9; 9, 18
84	9; 6, 18
85	9; 4, 1, 1, 4, 18
86	9; 3, 1, 1, 1, 8, 1, 1, 1, 3, 18
87	9; 3, 18
88	9; 2, 1, 1, 1, 2, 18
89	9; 2, 3, 3, 2, 18
90	9; 2, 18
91	9; 1, 1, 5, 1, 5, 1, 1, 18
92	9; 1, 1, 2, 4, 2, 1, 1, 18
93	9; 1, 1, 1, 4, 6, 4, 1, 1, 1, 18
94	9; 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18
95	9; 1, 2, 1, 18
96	9; 1, 3, 1, 18
97	9; 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18
98	9; 1, 8, 1, 18
99	9; 1, 18
101	10; 20

As the table shows, the periodic part ends with $2\lfloor \sqrt{n}\rfloor$.

Title	table of continued fractions of $\sqrt{n}$ for
Canonical name	TableOfContinuedFractionsOfsqrtnFor1N102
Date of creation	2013-03-22 17:30:25
Last modified on	2013-03-22 17:30:25
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Data Structure
Classification	msc 11A25