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Periodic Continued Fraction

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A periodic continued fraction is a continued fraction (generally a regular continued fraction) whose terms eventually repeat from some point onwards. The minimal number of repeating terms is called the period of the continued fraction. All nontrivial periodic continued fractions represent irrational numbers. In general, an infinite simple fraction (periodic or otherwise) represents a unique irrational number, and each irrational number has a unique infinite continued fraction.

The square root of a squarefree integer has a periodic continued fraction of the form

sqrt(n)=[a_0;a_1,a_2,a_3,...,a_2,a_1,2a_0^_]
(1)

(Rose 1994, p. 130), where the repeating portion (excluding the last term) is symmetric upon reversal, and the central term may appear either once or twice.

If D is not a square number, then the terms of the continued fraction of sqrt(n) satisfy

0<a_k<2sqrt(n).
(2)

An even stronger result is that a continued fraction is periodic iff it is a root of a quadratic polynomial. Calling the portion of a number x remaining after a given convergent the "tail," it must be true that the relationship between the number x and terms in its tail is of the form

x=(ax+b)/(cx+d),
(3)

which can only lead to a quadratic equation.

PeriodicContinuedFractionPeriods

The periods of the continued fractions of the square roots of the first few nonsquare integers 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, ... (OEIS A000037) are 1, 2, 1, 2, 4, 2, 1, 2, 2, 5, ... (OEIS A013943; Williams 1981, Jacobson et al. 1995). These numbers and their continued fraction representations are summarized in the following table.

Nalpha_(sqrt(N))Nalpha_(sqrt(N))
2[1,2^_]22[4,1,2,4,2,1,8^_]
3[1,1,2^_]23[4,1,3,1,8^_]
5[2,4^_]24[4,1,8^_]
6[2,2,4^_]26[5,10^_]
7[2,1,1,1,4^_]27[5,5,10^_]
8[2,1,4^_]28[5,3,2,3,10^_]
10[3,6^_]29[5,2,1,1,2,10^_]
11[3,3,6^_]30[5,2,10^_]
12[3,2,6^_]31[5,1,1,3,5,3,1,1,10^_]
13[3,1,1,1,1,6^_]32[5,1,1,1,10^_]
14[3,1,2,1,6^_]33[5,1,2,1,10^_]
15[3,1,6^_]34[5,1,4,1,10^_]
17[4,8^_]35[5,1,10^_]
18[4,4,8^_]37[6,12^_]
19[4,2,1,3,1,2,8^_]38[6,6,12^_]
20[4,2,8^_]39[6,4,12^_]
21[4,1,1,2,1,1,8^_]40[6,3,12^_]

An upper bound for the length of the period is roughly O(lnDsqrt(D)). The least positive ns such that the continued fraction of sqrt(n) has period p=1, 2, ... are 2, 3, 41, 7, 13, 19, 58, 31, 106, ... (OEIS A013646). The first few values of n such that the continued fraction of sqrt(n) has period p are summarized below for small p.

pOEISn
1A0025222, 5, 10, 17, 26, 37, 50, 65, 82, 101, ...
2A0136423, 6, 8, 11, 12, 15, 18, 20, 24, 27, ...
3A01364341, 130, 269, 370, 458, ...
4A0136447, 14, 23, 28, 32, 33, 34, 47, 55, 60, ...
5A01033713, 29, 53, 74, 85, 89, 125, 173, 185, 218, ...
6A02034719, 21, 22, 45, 52, 54, 57, 59, 70, 77, ...
7A01033858, 73, 202, 250, 274, 314, 349, 425, ...
8A02034831, 44, 69, 71, 91, 92, 108, 135, 153, 158, ...
9A010339106, 113, 137, 149, 265, 389, 493, ...
10A02034943, 67, 86, 93, 115, 116, 118, 129, 154, 159, ...

The values of n at which the period of the continued fraction of sqrt(n) increases are 1, 2, 3, 7, 13, 19, 31, 43, 46, 94, 139, 151, 166, 211, 331, 421, 526, 571, ... (OEIS A013645).

General identities for periodic continued fractions include

[a^_]=(a+sqrt(a^2+4))/2
(4)
[1,a^_]=(2-a+sqrt(a^2+4))/2
(5)
[a,2a^_]=sqrt(a^2+1)
(6)
[a,b^_]=(-ab+sqrt(ab(ab+4)))/(2a)
(7)
[a_1,...,a_n^_]=(-(q_(n-1)-p_n)+sqrt((q_(n-1)-p_n)^2+4q_np_(n-1)))/(2q_n)
(8)
[a_0,b_1,...,b_n^_]=a_0+1/([b_1,...,b_n^_])
(9)
[b_1,...,b_n^_]=([b_1,...,b_n^_]p_n+p_(n-1))/([b_1,...,b_n^_]q_n+q_(n-1))
(10)

(Wall 1948, pp. 39 and 83).

The first follows from

alpha=a+1/(a+1/(a+1/(a+...)))
(11)
=a+1/(a+(1/(a+1/(a+...)))).
(12)

Therefore,

alpha-a=1/(a+1/(a+1/(a+...))),
(13)

so plugging (13) into (12) gives

alpha=a+1/(a+(alpha-a))=a+1/alpha.
(14)

Expanding

alpha^2-aalpha-1=0,
(15)

and solving using the quadratic formula gives

alpha=(a+sqrt(a^2+4))/2.
(16)

The analog of this treatment in the general case gives

alpha=(alphap_n+p_(n-1))/(alphaq_n+q_(n-1)).
(17)

See also

Continued Fraction, Convergent, Near Noble Number, Noble Number, Quadratic Surd, Simple Continued Fraction

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References

Liberman, H. Simple Continued Fractions: An Elementary to Research Level Approach. SMD Stock Analysts, 2003.Rose, H. E. A Course in Number Theory, 2nd ed. Oxford, England: Oxford University Press, 1994.Rosen, K. H. Elementary Number Theory and Its Applications. New York: Addison-Wesley, p. 426, 1980.Sloane, N. J. A. Sequences A010337, A010338, A010339, A013642, A013643, A013644, A013645, A013646, A020347, A020348, and A020349 in "The On-Line Encyclopedia of Integer Sequences."Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.

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Periodic Continued Fraction

Cite this as:

Weisstein, Eric W. "Periodic Continued Fraction." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PeriodicContinuedFraction.html

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