TableOfContinuedFractionsOfSqrtnFor1N102 2007-08-26 17:59:48 2007-08-26 17:59:48 Data Structure square root % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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. \begin{tabular}{|r|l|} $n$ & 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 \\ \end{tabular} As the table shows, the periodic part ends with $2 \lfloor \sqrt{n} \rfloor$.