循环连分数 是一种可表示为以下形式的连分数 ：
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{\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {\ddots }{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {\ddots }{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}\,}
前k+1个部分分母不算，后面的部分分母[a k +1a k +2a k +m
2
{\displaystyle {\sqrt {2}}}
循环连分数的部份分母{a i
1770年，拉格朗日 证明一个数字能表示成循环连分数，若且唯若 此数为二次无理数 [ 1]
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{\displaystyle {\sqrt {3}}=1.732\ldots =[1;1,2,1,2,1,2,\ldots ]}
在此条目以下的内容会限制在部份分母为正整数的循环连分数。
因为循环连分数的分子都是1，因此可以用以下简化的方式记录循环连分数：
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{\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},a_{k+1},a_{k+2},\dots ,a_{k+m},a_{k+1},a_{k+2},\dots ,a_{k+m},\dots ]\\&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\overline {a_{k+1},a_{k+2},\dots ,a_{k+m}}}]\end{aligned}}}
第二行的括线 表示循环的部份。有些教材书会用以下的写法
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{\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\dot {a}}_{k+1},a_{k+2},\dots ,{\dot {a}}_{k+m}]\end{aligned}}}
循环部份的第一个数字和最后一个数字上方加上点识别。
若循环连分数中都是循环部份，没有不循环的第一部份，也就是k = -1, a0 = am ，则
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{\displaystyle x=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}],}
这样的循环连分数称为纯循环连分数（purely periodic）。例如黄金比例 φ的循环连分数是
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{\displaystyle [1;1,1,1,\dots ]}
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{\displaystyle {\sqrt {2}}}
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{\displaystyle [1;2,2,2,\dots ]}
循环连分数可以和实数的二次无理数 一一对应。其对应关系在明可夫斯基问号函数
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{\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}],}
此纯循环连分数可以写成
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{\displaystyle x={\frac {\alpha x+\beta }{\gamma x+\delta }}}
其中
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{\displaystyle \alpha ,\beta ,\gamma ,\delta }
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{\displaystyle \alpha \delta -\beta \gamma =1.}
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{\displaystyle S={\begin{pmatrix}1&0\\1&1\end{pmatrix}}}
表示移位，因此
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{\displaystyle S^{n}={\begin{pmatrix}1&0\\n&1\end{pmatrix}}}
以下这个类似反射
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{\displaystyle T\mapsto {\begin{pmatrix}-1&1\\0&1\end{pmatrix}}}
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{\displaystyle T^{2}=I}
单位模矩阵
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{\displaystyle S^{a_{1}}TS^{a_{2}}T\cdots TS^{a_{m}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}}
而
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{\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}]={\frac {\alpha x+\beta }{\gamma x+\delta }}}
是其显式式。因为所有的矩阵元素都是整数，矩阵也属于模群
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{\displaystyle SL(2,\mathbb {Z} ).}
^ Kenneth H. Rosen. Elementary Number Theory and Its Applications.
Long, Calvin T. Elementary Introduction to Number Theory 3 Sub. Waveland Pr Inc. 1972. LCCN 77-171950 Pettofrezzo, Anthony Joseph; Byrkit, Donald R. Elements of Number Theory 11. Englewood Cliffs: Prentice Hall. 1970. ISBN 9780132683005LCCN 77-81766 Khinchin, A. Ya. Continued Fractions ISBN 0-486-69630-8 
![{\displaystyle {\sqrt {3}}=1.732\ldots =[1;1,2,1,2,1,2,\ldots ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64502475b8b77c8d93447a3e0d9e95b1d642da4e)
![{\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},a_{k+1},a_{k+2},\dots ,a_{k+m},a_{k+1},a_{k+2},\dots ,a_{k+m},\dots ]\\&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\overline {a_{k+1},a_{k+2},\dots ,a_{k+m}}}]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06b6d3cec2c6ddf1be04eb487576eb5653b21260)
![{\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\dot {a}}_{k+1},a_{k+2},\dots ,{\dot {a}}_{k+m}]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/056cfbee8f961adb15a057e894a6dc3e5541c407)
![{\displaystyle x=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b04729c5e6ae3c7807bffa2397b34a817674df23)
![{\displaystyle [1;1,1,1,\dots ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dafd8bdba741b261bc080fe2d9fcc990e3f16d3c)
![{\displaystyle [1;2,2,2,\dots ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62bc4adab7fb847a0cd75dd7ec00888296172dba)
![{\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cd84ced002ba3a12e9c7e56caa3c05c160bb93)
![{\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}]={\frac {\alpha x+\beta }{\gamma x+\delta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5681df078e1ca6851fbcb88653eead755d0369df)
. University of Chicago Press. 1964 [Originally published in Russian, 1935]. ISBN 0-486-69630-8. (This is now available as a reprint from Dover Publications.)
