√2 の連分数展開を求めてみる(展開の方法は >>41 参照)。

√2 = 1 + (√2 - 1)
1/(√2 - 1) = √2 + 1 = 2 + (√2 - 1)

よって
√2 = [1, 2, 2, . . . ]

同様に

√3 = 1 + (√3 - 1)
1/(√3 - 1) = (√3 + 1)/2 = 1 + (√3 - 1)/2
2/(√3 - 1) = √3 + 1 = 2 + (√3 - 1)

よって
√3 = [1, 1, 2, 1, 2, . . . ]


√5 = 2 + (√5 - 2)
1/(√5 - 2) = √5 + 2 = 4 + (√5 - 2)
√5 = [2, 4, 4, 4. . . ]

√7 = 2 + (√7 - 2)
1/(√7 - 2) = (√7 + 2)/3 = 1 + (√7 - 1)/3
3/(√7 - 1) = (√7 + 1)/2 = 1 + (√7 - 1)/2
2/(√7 - 1) = (√7 + 1)/3 = 1 + (√7 - 2)/3
3/(√7 - 2) = √7 + 2 = 4 + (√7 - 2)
√7 = [2, 1, 1, 1, 4, 1, 1, 1, 4, . . . ]