Periodic continued fraction

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

$x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{\ddots}{\quad\ddots\quad a_k + \cfrac{1}{a_{k+1} + \cfrac{\ddots}{\quad\ddots\quad a_{k+m-1} + \cfrac{1}{a_{k+m} + \cfrac{1}{a_{k+1} + \cfrac{1}{a_{k+2} + {\ddots}}}}}}}}}\,$

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

The partial denominators {a_i} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of simple continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators a_i (i ≥ 1) are positive integers.

Purely periodic and periodic fractions

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

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

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

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

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

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

the regular continued fraction x is said to be purely periodic. For example, the regular continued fraction for the golden ratio φ – given by [1; 1, 1, 1, …] – is purely periodic, while the regular continued fraction for the square root of two – [1; 2, 2, 2, …] – is periodic, but not purely periodic.

Relation to quadratic irrationals

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

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

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

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

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.^[3] 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.

Reduced surds

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

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

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

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

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.

Length of the repeating block

By analyzing the sequence of combinations

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

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^[4][5] based on the divisor function have shown that L(D), the length of the repeating block for a quadratic surd of discriminant D, is given by

$L(D)={\mathcal {O}}({\sqrt {D}}\ln {D})$

where the big O means "on the order of", or "asymptotically proportional to" (see big O notation).

See also

• Hermite's problem
• Continued fraction method of computing square roots
• Restricted partial quotients

Notes

1. Pettofrezzo & Byrkit (1970, p. 158)
2. Long (1972, p. 187)
3. Davenport, H. (1982). "The Higher Arithmetic". Cambridge University Press: 104. ISBN 0-521-28678-6.
4. Hickerson, Dean R. (1973). "Length of period of simple continued fraction expansion of √d". Pacific J. Math. 46: 429–432. doi:10.2140/pjm.1973.46.429.
5. Podsypanin, E.V. (1982). "Length of the period of a quadratic irrational". Journal of Soviet Mathematics. 18 (6): 919–923. doi:10.1007/BF01763963.

References

• Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950
• Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766
• Rockett, Andrew M.; Szüsz, Peter (1992). Continued Fractions. World Scientific. ISBN 981-02-1052-3.