这个连分数展开有什么规律?

mathematica

1. ContinuedFraction[Sqrt[2] + Sqrt[3], 2000]

复制代码

结果
{3, 6, 1, 5, 7, 1, 1, 4, 1, 38, 43, 1, 3, 2, 1, 1, 1, 1, 2, 4, 1, 4, \
5, 1, 5, 1, 7, 22, 2, 5, 1, 1, 2, 1, 1, 31, 2, 1, 1, 3, 1, 44, 1, 89, \
1, 8, 5, 2, 5, 1, 1, 4, 2, 8, 1, 17, 1, 4, 3, 4, 3, 2, 1, 1, 4, 2, 1, \
9, 1, 15, 13, 1, 39, 20, 2, 152, 3, 2, 4, 1, 30, 1, 3, 1, 2, 1, 2, \
16, 3, 24, 1, 9, 1, 172, 3, 1, 1, 1, 1, 27, 1, 1, 1, 8, 1, 1, 2, 8, \
2, 4, 1, 1, 1, 5, 61, 1, 2, 2, 6, 1, 4, 1, 1, 7, 1, 1, 1, 2, 7, 2, \
24, 3, 1, 2, 2, 9, 2, 9, 1, 8, 1, 3, 30, 1, 3, 1, 2, 3, 1, 7, 1, 4, \
4, 1, 1, 1, 103, 1, 1, 2, 1, 3, 1, 4, 2, 1, 1, 4, 1, 5, 2, 1, 3, 1, \
5, 1, 12, 1, 1, 3, 1, 3, 1, 3, 1, 1, 6, 1, 2, 1, 3, 1, 1, 3, 3, 2, 1, \
2, 21, 1, 2, 2, 1, 2, 2, 2, 13, 1, 1, 8, 2, 2, 4, 1, 11, 1, 1, 159, \
4, 2, 1, 1, 1, 2, 8, 6, 1, 4, 26, 2, 1, 2, 1, 17, 1, 1, 8, 1, 158, 1, \
1, 1, 2, 1, 3, 19, 7, 1, 1, 3, 2, 2, 1, 8, 3, 2, 4, 2, 2, 1, 1, 13, \
1, 9, 37, 2, 15, 4, 17, 69, 1, 6, 9, 2, 3, 12, 1, 7, 1, 1, 2, 2, 3, \
3, 8, 2, 2, 1, 2, 70, 131, 1, 13, 1, 5, 1, 2, 3, 1, 18, 29, 1, 1, 1, \
13, 1, 12, 1, 1, 2, 2, 2, 2, 1, 135, 12, 16, 1, 9, 72, 1, 10, 1, 7, \
1, 1, 1, 1, 11, 153, 2, 5, 6, 1, 1, 16, 1, 1342, 1, 3, 1, 1, 1, 3, 1, \
1, 1, 1, 4, 5, 6, 1, 1, 1, 151, 5, 2, 2, 5, 5, 1, 9, 7, 1, 1, 24, 1, \
2, 2, 1, 3, 1, 7, 1, 4, 1, 1, 3, 2, 1, 2, 9, 3, 1, 1, 20, 4, 23, 2, \
1, 1, 1, 1, 1, 5, 4, 1, 6, 2, 1, 2, 1, 1, 1, 1, 61, 1, 1, 2, 2, 2222, \
1, 8, 1, 1, 2, 8, 9, 2, 2, 1, 1, 9, 1, 86, 2, 6, 2, 24, 3, 1, 1, 4, \
2, 3, 2, 2, 1, 7, 1, 3, 3, 1, 1, 1, 6, 1, 1, 1, 3, 14, 1, 1, 3, 1, 6, \
58, 1, 1, 2, 1, 1, 1, 3, 3, 3, 2, 3, 1, 4, 2, 2, 2, 159, 5, 1, 1, 28, \
3, 2, 1, 3, 1, 5, 4, 6, 1, 4, 1, 7, 1, 1, 1, 6, 2, 4, 3, 1, 1, 1, 1, \
1, 1, 17, 7, 3, 1, 10, 3, 3, 2, 255, 11, 5, 5, 2, 2, 26, 1, 3, 12, 1, \
3, 2, 1, 2, 1, 1, 2, 16, 1, 6, 1, 1, 1, 31, 1, 29, 2, 1, 1, 51, 1, 1, \
2, 1, 9, 1, 2, 1, 2, 115, 1, 4, 2, 4, 1, 1, 1, 1, 2, 5, 1, 3, 3, 1, \
1, 3, 1, 10, 1, 1, 2, 49, 2, 2, 1, 6, 1, 13, 106, 1, 2, 8, 3, 141, 2, \
29, 1, 35, 18, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 7, 2, 5, 5, 1, 1, \
12, 2, 19, 1, 1, 4, 1, 19, 10, 4, 1, 1, 2, 23, 1, 4, 12, 12, 12, 4, \
1, 3, 1, 3, 1, 3, 7, 13, 1, 2, 4, 10, 11, 1, 1, 1, 8, 2, 1, 1, 47, 2, \
4, 1, 11, 2, 1, 4, 1, 7, 1, 5, 2, 2, 2, 2, 17, 1, 6, 1, 1, 1, 6, 3, \
1, 1, 1, 6, 3, 1, 3, 2, 1, 2, 1, 1, 9, 1, 1, 1, 6, 4, 1, 6, 1, 6, 1, \
1, 1, 1, 245, 3, 13, 1, 1, 2, 4, 2, 2, 1, 1, 2, 1, 1, 1, 1, 6, 1, 7, \
3, 2, 1, 10, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 15, 1, 10, 36, 2, 1, 44, \
1, 1, 1, 5, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 3, 6, 1, 4, 2, 1, 2, 5, 6, \
1, 1, 1, 3, 1, 4, 5, 1, 1, 1, 4, 3, 4, 3, 1, 12, 1, 1, 1, 1, 1, 1, 2, \
2, 1, 6, 2, 3, 1, 3, 2, 4, 2, 49, 3, 1, 12, 1, 3, 11, 7, 2, 1, 5, 6, \
2, 1, 1, 1, 2, 4, 2, 1, 1, 21, 43, 1, 3, 2, 2, 1, 9, 42, 8, 3, 2, 4, \
1, 1228, 1, 28, 82, 10, 3, 1, 1, 1, 8, 1, 9, 1, 1, 23, 1, 1, 1, 1, 6, \
2, 2, 4, 1, 2, 7, 1, 8, 2, 3, 4, 2, 1, 1, 1, 2, 9, 1, 5, 143, 1, 2, \
10, 1, 3, 1, 1, 1, 1, 3, 1, 20, 1, 1, 240, 1, 44, 2, 1, 36, 4, 1, 1, \
2, 1, 1, 1, 3, 3, 4, 5, 1, 1, 3, 1, 3, 1, 3, 2, 4, 7, 59, 4, 1, 1, 1, \
7, 2, 1, 4, 1, 1, 4, 1, 9, 1, 5, 1, 1, 9, 1, 7, 1, 3, 5, 530, 4, 2, \
2, 6, 2, 5, 1, 20, 2, 6, 34, 3, 2, 6, 2, 3, 1, 2, 1, 1, 4, 45, 3, 1, \
31, 1, 8, 1, 382, 4, 1, 3, 2, 3, 1, 6, 7, 3, 1, 1, 1, 1, 1, 1, 2, 2, \
2, 1, 1, 1, 4, 1, 1, 1, 1, 16, 4, 3, 2, 1, 3, 1, 18, 1, 1, 3, 5, 2, \
3, 29, 3, 1, 1, 1, 1, 2, 4, 1, 5, 1, 4, 10, 5, 1, 17, 1, 5, 1, 3, 11, \
2, 1, 23, 2, 1, 7, 1, 1, 3, 2, 18, 2, 14, 3, 1, 5, 1, 2, 6, 2, 2, 2, \
5, 4, 1, 2, 23, 16, 3, 3, 8, 1, 4, 1, 1, 1, 1, 14, 3, 2, 6, 1, 24, 5, \
1, 3, 2, 2, 2, 1, 1, 7, 1, 1, 17, 2, 1, 1, 9, 1, 5, 1, 1, 1, 9, 38, \
1, 3, 7, 1, 1, 1, 1, 2, 15, 3, 1, 1, 1, 21, 2, 3, 19, 4, 316, 1, 3, \
7, 11, 3, 1, 5, 2, 5, 5, 1, 1, 2, 3, 2, 1, 1, 2, 1, 1, 11, 5, 1, 16, \
2, 1, 7, 1, 1, 8, 13, 9, 13, 1, 26, 3, 4, 3, 1, 1, 5, 3, 16, 2, 1, 1, \
3, 1, 3, 1, 33, 1, 1, 5, 1, 6, 3, 1, 2, 1, 1, 1, 6, 1, 1, 8, 2, 2, \
52, 1, 1, 1, 3, 1, 1, 1, 1, 1, 10, 1, 1, 1, 5, 1, 10, 1, 20, 4, 1, 1, \
6, 13, 3, 2, 37, 1, 13, 1, 37, 2, 1, 6, 1, 2, 1, 1, 1, 1, 16, 2, 1, \
1, 1, 7, 7, 3, 4, 2, 1, 4, 1, 1, 6, 3, 9, 21, 1, 5, 1, 2, 1, 1, 5, 1, \
12, 30, 9, 1, 6, 6, 3, 1, 3, 1, 1, 3, 3, 1, 4, 4, 2, 12, 1, 2, 1, 6, \
1, 8, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 2, 1, \
1, 2, 4, 1, 1, 7, 2, 1, 1, 2, 14, 1, 1, 13, 1, 2, 2, 1, 3, 6, 1, 3, \
2, 1, 1, 1, 1, 1, 4, 39, 2, 1, 1, 3, 9, 2, 1, 5, 1, 21, 1, 6, 7, 1, \
2, 1, 2, 1, 6, 3, 8, 1, 1, 1, 1, 2, 1, 4, 16, 5, 1, 1, 1, 8, 2, 1, 7, \
1, 7, 1, 3, 1, 3, 4, 2, 1, 1, 2, 2, 1, 2, 5, 1, 3, 1, 1, 1, 5, 2, 1, \
21, 1, 2, 5, 1, 2, 3, 3, 2, 2, 3, 1, 5, 2, 1, 1, 1, 9, 83, 3, 7, 5, \
3, 1, 28, 1, 3, 1, 5, 1, 2, 6, 3, 1, 1, 5, 4, 11, 4, 1, 1, 1, 1, 5, \
1, 38, 2, 6, 3, 1, 1, 1, 1, 7, 1, 2, 236, 13, 1, 3, 5, 1, 17, 1, 1, \
4, 2, 2, 2, 2, 4, 2, 1, 9, 50, 1, 10, 1, 1, 1, 2, 36, 2, 1, 8, 6, 1,
  36, 3, 1, 1, 43, 1, 3, 4, 1, 6, 3, 2, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, \
7, 1, 3, 1, 7, 1, 5, 5, 1, 80, 1, 6, 7, 1, 4, 6, 1, 431, 1, 11, 1, 4, \
6, 9, 1, 3, 12, 2, 7, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 4, 2, 11, \
5, 1, 3, 1, 3, 3, 1, 1, 7, 1, 6, 1, 1, 5, 4, 47, 5, 3, 29, 1, 3, 4, \
16, 1, 1, 1, 4, 3, 31, 1, 2, 1, 7, 1, 2, 3, 1, 3, 2, 65, 1, 94, 2, 3, \
5, 12, 3, 3, 5, 2, 4, 13, 1, 5, 1, 1, 1, 1, 2, 5, 1, 1, 6, 9, 4, 1, \
229, 1, 7, 1, 1, 2, 1, 22, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 15, 2, 1, \
30, 1, 6, 16, 1, 3, 1, 1, 7, 1, 1, 4, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, \
1, 3, 1, 1, 2, 1, 20, 12, 9, 1, 115, 4, 1, 7, 1, 6, 2, 50, 1, 3, 2, \
2, 2, 28, 1, 11, 1, 1, 2, 1, 3, 3, 29, 8, 2, 2, 1, 1, 2, 31, 1, 2, 1, \
18, 1, 1, 1, 1, 4, 1, 1, 1, 2, 27, 1, 1, 1, 1, 1, 1, 2, 2, 1, 9, 9, \
2, 1, 8, 1, 2, 92, 3, 1, 1, 1, 3, 1, 7, 1, 3, 2, 1, 14, 1, 1, 1, 25, \
1, 1, 2, 2, 5, 42, 1, 1, 4, 6, 1, 1, 1, 2, 18, 2, 1, 7, 2, 2, 3, 1, \
15, 1, 1, 20, 2, 1, 1, 3, 3, 1, 2, 2, 32, 1, 5, 1, 27, 18, 3, 19, 3,
  3, 3, 3, 1, 1, 1, 1, 5, 5, 16, 1, 1, 9, 1, 4, 1, 1, 1, 1, 4, 10, 1, \
2, 9, 26, 30, 1, 7, 3, 42, 1, 2, 1, 1, 15, 1, 3, 2, 1, 1, 1, 5, 6, 1, \
11, 12, 11, 3, 7, 4, 2, 2, 1, 1, 5, 4, 3, 3, 1, 1, 3, 1, 1, 1, 1, 4, \
1, 6, 6, 1, 8, 103, 1, 1, 1, 1, 4, 4, 1, 3, 2, 2, 15, 1, 1, 1, 2, 15, \
1, 1, 2, 1, 2, 6, 1, 1, 27, 1, 3, 1, 9, 1, 1, 1, 1, 1, 1, 10, 1, 2, \
2, 4, 2, 2, 7, 7, 1, 1, 5, 1, 1, 71, 1, 4, 2, 1, 1, 1, 10, 2, 1, 1, \
2, 1, 3, 4, 1, 3, 5, 4, 10, 7, 1, 3, 2, 3, 1, 1, 2, 10, 1, 2, 2, 1, \
1, 1, 5, 8, 1, 1, 3, 3, 38, 2, 2, 2, 4, 12, 2, 1, 4, 1, 4, 1, 5, 6, \
2, 1, 3, 4, 3, 1, 15, 270, 7, 5, 20, 1, 1, 5, 44, 1, 44, 1, 2, 1, 4, \
3, 1, 3, 9, 5, 2, 1, 3, 1, 1, 3, 1, 5, 1, 12, 3, 3, 2, 1, 3, 5, 3, 1, \
5, 3, 1, 9, 3, 1, 9, 1}
统计一下
Tally[%]
结果
{{3, 202}, {6, 62}, {1, 858}, {5, 87}, {7, 54}, {4, 107}, {38,
  4}, {43, 3}, {2, 312}, {22, 2}, {31, 5}, {44, 5}, {89, 1}, {8,
  29}, {17, 8}, {9, 37}, {15, 10}, {13, 15}, {39, 2}, {20, 8}, {152,
  1}, {30, 5}, {16, 13}, {24, 5}, {172, 1}, {27, 4}, {61, 2}, {103,
  2}, {12, 19}, {21, 6}, {11, 15}, {159, 2}, {26, 4}, {158, 1}, {19,
  5}, {37, 3}, {69, 1}, {70, 1}, {131, 1}, {18, 7}, {29, 6}, {135,
  1}, {72, 1}, {10, 18}, {153, 1}, {1342, 1}, {151, 1}, {23,
  5}, {2222, 1}, {86, 1}, {14, 5}, {58, 1}, {28, 4}, {255, 1}, {51,
  1}, {115, 2}, {49, 2}, {106, 1}, {141, 1}, {35, 1}, {47, 2}, {245,
  1}, {36, 4}, {42, 3}, {1228, 1}, {82, 1}, {143, 1}, {240, 1}, {59,
  1}, {530, 1}, {34, 1}, {45, 1}, {382, 1}, {316, 1}, {33, 1}, {52,
  1}, {83, 1}, {236, 1}, {50, 2}, {80, 1}, {431, 1}, {65, 1}, {94,
  1}, {229, 1}, {92, 1}, {25, 1}, {32, 1}, {71, 1}, {270, 1}}
还是看不出啥规律
换成
ContinuedFraction[Sqrt[163], 200]
{12, 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, 1, 2, 3, 3, 1, 24, 1, 3, 3, \
2, 1, 1, 7, 1, 11, 1, 7, 1, 1, 2, 3, 3, 1, 24, 1, 3, 3, 2, 1, 1, 7, \
1, 11, 1, 7, 1, 1, 2, 3, 3, 1, 24, 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, \
1, 1, 2, 3, 3, 1, 24, 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, 1, 2, 3, \
3, 1, 24, 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, 1, 2, 3, 3, 1, 24, 1, \
3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, 1, 2, 3, 3, 1, 24, 1, 3, 3, 2, 1, \
1, 7, 1, 11, 1, 7, 1, 1, 2, 3, 3, 1, 24, 1, 3, 3, 2, 1, 1, 7, 1, 11, \
1, 7, 1, 1, 2, 3, 3, 1, 24, 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, 1, \
2, 3, 3, 1, 24, 1, 3, 3, 2, 1, 1, 7, 1, 11, 1, 7, 1, 1, 2, 3, 3, 1, \
24, 1}

Tally[%]

{{12, 1}, {1, 89}, {3, 44}, {2, 22}, {7, 22}, {11, 11}, {24, 11}}

这个相对还是能看出规律的

mathematica

1. Tally[ContinuedFraction[Sqrt[2] + Sqrt[3], 200]]

复制代码

结果
{{3, 21}, {6, 3}, {1, 87}, {5, 9}, {7, 5}, {4, 14}, {38, 1}, {43,
  1}, {2, 29}, {22, 1}, {31, 1}, {44, 1}, {89, 1}, {8, 5}, {17,
  1}, {9, 4}, {15, 1}, {13, 1}, {39, 1}, {20, 1}, {152, 1}, {30,
  2}, {16, 1}, {24, 2}, {172, 1}, {27, 1}, {61, 1}, {103, 1}, {12,
  1}, {21, 1}}

1. Tally[ContinuedFraction[Sqrt[2] + Sqrt[3], 2000]]

复制代码

{{3, 202}, {6, 62}, {1, 858}, {5, 87}, {7, 54}, {4, 107}, {38,
  4}, {43, 3}, {2, 312}, {22, 2}, {31, 5}, {44, 5}, {89, 1}, {8,
  29}, {17, 8}, {9, 37}, {15, 10}, {13, 15}, {39, 2}, {20, 8}, {152,
  1}, {30, 5}, {16, 13}, {24, 5}, {172, 1}, {27, 4}, {61, 2}, {103,
  2}, {12, 19}, {21, 6}, {11, 15}, {159, 2}, {26, 4}, {158, 1}, {19,
  5}, {37, 3}, {69, 1}, {70, 1}, {131, 1}, {18, 7}, {29, 6}, {135,
  1}, {72, 1}, {10, 18}, {153, 1}, {1342, 1}, {151, 1}, {23,
  5}, {2222, 1}, {86, 1}, {14, 5}, {58, 1}, {28, 4}, {255, 1}, {51,
  1}, {115, 2}, {49, 2}, {106, 1}, {141, 1}, {35, 1}, {47, 2}, {245,
  1}, {36, 4}, {42, 3}, {1228, 1}, {82, 1}, {143, 1}, {240, 1}, {59,
  1}, {530, 1}, {34, 1}, {45, 1}, {382, 1}, {316, 1}, {33, 1}, {52,
  1}, {83, 1}, {236, 1}, {50, 2}, {80, 1}, {431, 1}, {65, 1}, {94,
  1}, {229, 1}, {92, 1}, {25, 1}, {32, 1}, {71, 1}, {270, 1}}

1. Tally[ContinuedFraction[Sqrt[163], 200]]

复制代码

{{12, 1}, {1, 89}, {3, 44}, {2, 22}, {7, 22}, {11, 11}, {24, 11}}

1. Tally[ContinuedFraction[Sqrt[163], 2000]]

复制代码

{{12, 1}, {1, 889}, {3, 444}, {2, 222}, {7, 222}, {11, 111}, {24, 111}}
通过展开不同项数目,然后统计发现$\sqrt{163}$是有规律的,
但是$\sqrt{2}+\sqrt{3}$有什么规律呢?