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Questions in category: 初等数论 (Elementary Number Theory).

Farey 序列

Posted by haifeng on 2011-08-10 14:55:36 last update 2015-04-30 15:26:54 | Answers (0) | 收藏

$n$ 阶的 Farey 序列是指下述集合

\[ \biggl\{\frac{p}{q}\mid (p,q)=1,1\leqslant p,q\leqslant n\biggr\}\cup\biggr\{\frac{0}{1}\biggr\} \]

按大小排列成的一个序列. 如

\[ F_6=\biggl\{\frac{0}{1},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{1}{1}\biggr\}\]

\[ F_7=\biggl\{\frac{0}{1},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{1}{2},\frac{4}{7},\frac{3}{5},\frac{2}{3},\frac{5}{7},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{1}{1}\biggr\} \]

\[ F_{8}=\biggl\{\frac{0}{8},\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{3}{8},\frac{2}{5},\frac{3}{7},\frac{1}{2},\frac{4}{7},\frac{3}{5},\frac{5}{8},\frac{2}{3},\frac{5}{7},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8},\frac{1}{1}\biggr\} \]

\[ F_{9}=\biggl\{\frac{0}{9},\frac{1}{9},\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{2}{9},\frac{1}{4},\frac{2}{7},\frac{1}{3},\frac{3}{8},\frac{2}{5},\frac{3}{7},\frac{4}{9},\frac{1}{2},\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{5}{8},\frac{2}{3},\frac{5}{7},\frac{3}{4},\frac{7}{9},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8},\frac{8}{9},\frac{1}{1}\biggr\} \]

\[ F_{10}=\biggl\{\frac{0}{10},\frac{1}{10},\frac{1}{9},\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{1}{5},\frac{2}{9},\frac{1}{4},\frac{2}{7},\frac{3}{10},\frac{1}{3},\frac{3}{8},\frac{2}{5},\frac{3}{7},\frac{4}{9},\frac{1}{2},\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{5}{8},\frac{2}{3},\frac{7}{10},\frac{5}{7},\frac{3}{4},\frac{7}{9},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8},\frac{8}{9},\frac{9}{10},\frac{1}{1}\biggr\} \]

\[ F_{11}=\biggl\{\frac{0}{11},\frac{1}{11},\frac{1}{10},\frac{1}{9},\frac{1}{8},\frac{1}{7},\frac{1}{6},\frac{2}{11},\frac{1}{5},\frac{2}{9},\frac{1}{4},\frac{3}{11},\frac{2}{7},\frac{3}{10},\frac{1}{3},\frac{4}{11},\frac{3}{8},\frac{2}{5},\frac{3}{7},\frac{4}{9},\frac{5}{11},\frac{1}{2},\frac{6}{11},\frac{5}{9},\frac{4}{7},\frac{3}{5},\frac{5}{8},\frac{7}{11},\frac{2}{3},\frac{7}{10},\frac{5}{7},\frac{8}{11},\frac{3}{4},\frac{7}{9},\frac{4}{5},\frac{9}{11},\frac{5}{6},\frac{6}{7},\frac{7}{8},\frac{8}{9},\frac{9}{10},\frac{10}{11},\frac{1}{1}\biggr\} \]