In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form

${\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {1}{a_{k+1}+{\cfrac {\ddots }{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}}$

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where the initial block [a₀, a₁,...a_k] of k+1 partial denominators is followed by a block [a_k+1, a_k+2,...a_k+m] of m partial denominators that repeats ad infinitum. For example, ${\displaystyle {\sqrt {2}}}$ can be expanded to the periodic continued fraction [1,2,2,2,...].

This article considers only the case of periodic regular continued fractions. In other words, the remainder of this article assumes that all the partial denominators a_i (i ≥ 1) are positive integers. The general case, where the partial denominators a_i are arbitrary real or complex numbers, is treated in the article convergence problem.

Since all the partial numerators in a regular continued fraction are equal to unity we can adopt a shorthand notation in which the continued fraction shown above is written as

${\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},a_{k+1},a_{k+2},\dots ,a_{k+m},a_{k+1},a_{k+2},\dots ,a_{k+m},\dots ]\\&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\overline {a_{k+1},a_{k+2},\dots ,a_{k+m}}}]\end{aligned}}}$

where, in the second line, a vinculum marks the repeating block.^[1] Some textbooks use the notation

${\displaystyle {\begin{aligned}x&=[a_{0};a_{1},a_{2},\dots ,a_{k},{\dot {a}}_{k+1},a_{k+2},\dots ,{\dot {a}}_{k+m}]\end{aligned}}}$

where the repeating block is indicated by dots over its first and last terms.^[2]

If the initial non-repeating block is not present – that is, if k = -1, a₀ = a_m and

${\displaystyle x=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}],}$

the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction [1; 1, 1, 1, ...] of the golden ratio φ is purely periodic, while the regular continued fraction [1; 2, 2, 2, ...] of ${\displaystyle {\sqrt {2}}}$ is periodic, but not purely periodic.

Periodic continued fractions are in one-to-one correspondence with the real quadratic irrationals. The correspondence is explicitly provided by Minkowski's question-mark function. That article also reviews tools that make it easy to work with such continued fractions. Consider first the purely periodic part

${\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}],}$

This can, in fact, be written as

${\displaystyle x={\frac {\alpha x+\beta }{\gamma x+\delta }}}$

with the ${\displaystyle \alpha ,\beta ,\gamma ,\delta }$ being integers, and satisfying ${\displaystyle \alpha \delta -\beta \gamma =1.}$ Explicit values can be obtained by writing

${\displaystyle S={\begin{pmatrix}1&0\\1&1\end{pmatrix}}}$

which is termed a "shift", so that

${\displaystyle S^{n}={\begin{pmatrix}1&0\\n&1\end{pmatrix}}}$

and similarly a reflection, given by

${\displaystyle T\mapsto {\begin{pmatrix}-1&1\\0&1\end{pmatrix}}}$

so that ${\displaystyle T^{2}=I}$. Both of these matrices are unimodular, arbitrary products remain unimodular. Then, given ${\displaystyle x}$ as above, the corresponding matrix is of the form ^[3]

${\displaystyle S^{a_{1}}TS^{a_{2}}T\cdots TS^{a_{m}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}}$

and one has

${\displaystyle x=[0;{\overline {a_{1},a_{2},\dots ,a_{m}}}]={\frac {\alpha x+\beta }{\gamma x+\delta }}}$

as the explicit form. As all of the matrix entries are integers, this matrix belongs to the modular group ${\displaystyle SL(2,\mathbb {Z} ).}$

A quadratic irrational number is an irrational real root of the quadratic equation

${\displaystyle ax^{2}+bx+c=0}$

where the coefficients a, b, and c are integers, and the discriminant, b² − 4ac, is greater than zero. By the quadratic formula, every quadratic irrational can be written in the form

${\displaystyle \zeta ={\frac {P+{\sqrt {D}}}{Q}}}$

where P, D, and Q are integers, D > 0 is not a perfect square (but not necessarily square-free), and Q divides the quantity P² − D (for example (6+√8)/4). Such a quadratic irrational may also be written in another form with a square-root of a square-free number (for example (3+√2)/2) as explained for quadratic irrationals.

By considering the complete quotients of periodic continued fractions, Euler was able to prove that if x is a regular periodic continued fraction, then x is a quadratic irrational number. The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy.

Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic.^[4] Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another. Since there are only finitely many of these equations (the coefficients are bounded), the complete quotients (and also the partial denominators) in the regular continued fraction that represents x must eventually repeat.

The quadratic surd ${\displaystyle \zeta ={\frac {P+{\sqrt {D}}}{Q}}}$ is said to be reduced if ${\displaystyle \zeta >1}$ and its conjugate ${\displaystyle \eta ={\frac {P-{\sqrt {D}}}{Q}}}$ satisfies the inequalities ${\displaystyle -1<\eta <0}$. For instance, the golden ratio ${\displaystyle \phi =(1+{\sqrt {5}})/2=1.618033...}$ is a reduced surd because it is greater than one and its conjugate ${\displaystyle (1-{\sqrt {5}})/2=-0.618033...}$ is greater than −1 and less than zero. On the other hand, the square root of two ${\displaystyle {\sqrt {2}}=(0+{\sqrt {8}})/2}$ is greater than one but is not a reduced surd because its conjugate ${\displaystyle -{\sqrt {2}}=(0-{\sqrt {8}})/2}$ is less than −1.

Galois proved that the regular continued fraction which represents a quadratic surd ζ is purely periodic if and only if ζ is a reduced surd. In fact, Galois showed more than this. He also proved that if ζ is a reduced quadratic surd and η is its conjugate, then the continued fractions for ζ and for (−1/η) are both purely periodic, and the repeating block in one of those continued fractions is the mirror image of the repeating block in the other. In symbols we have

${\displaystyle {\begin{aligned}\zeta &=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}]\\[3pt]{\frac {-1}{\eta }}&=[{\overline {a_{m-1};a_{m-2},a_{m-3},\dots ,a_{0}}}]\,\end{aligned}}}$

where ζ is any reduced quadratic surd, and η is its conjugate.

From these two theorems of Galois a result already known to Lagrange can be deduced. If r > 1 is a rational number that is not a perfect square, then

${\displaystyle {\sqrt {r}}=[a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}].}$

In particular, if n is any non-square positive integer, the regular continued fraction expansion of √n contains a repeating block of length m, in which the first m − 1 partial denominators form a palindromic string.

By analyzing the sequence of combinations

${\displaystyle {\frac {P_{n}+{\sqrt {D}}}{Q_{n}}}}$

that can possibly arise when ζ = (P + √D)/Q is expanded as a regular continued fraction, Lagrange showed that the largest partial denominator a_i in the expansion is less than 2√D, and that the length of the repeating block is less than 2D.

More recently, sharper arguments ^[5][6][7] based on the divisor function have shown that the length of the repeating block for a quadratic surd of discriminant D is on the order of ${\displaystyle {\mathcal {O}}({\sqrt {D}}\ln {D}).}$

• Continued fraction – Number represented as a0+1/(a1+1/...)
• Generalized continued fraction – generalization of continued fractions in which the partial numerators and partial denominators can assume arbitrary complex valuesPages displaying wikidata descriptions as a fallback
• Hermite's problem
• Continued fraction method of computing square roots – Algorithms for calculating square roots
• Restricted partial quotients
• Continued fraction factorization – an integer factorization algorithmPages displaying wikidata descriptions as a fallback