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Hometable of continued fractions of $\sqrt{n}$ for $1 < n < 102$

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# table of continued fractions of $\sqrt{n}$ for $1<n<102$

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 $2\lfloor\sqrt{n}\rfloor$.

Major Section:

Reference

Type of Math Object:

Data Structure

Parent:

## Mathematics Subject Classification

11A25*no label found*

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## Comments

## "End" of periodic part of continued fractions of square root...

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

Interesting. So the continued fraction of sqrt(n^2 + 1) is always of the form n; 2n. Why is this?