# table of continued fractions of $\sqrt{n}$ for $1

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

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

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 $1 TableOfContinuedFractionsOfsqrtnFor1N102 2013-03-22 17:30:25 2013-03-22 17:30:25 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Data Structure msc 11A25