Deriving the Continued Fraction for Square Root of 2
Periodic Continued Fraction
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^_]](https://mathworld.wolfram.com/images/equations/PeriodicContinuedFraction/NumberedEquation1.svg) | (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 
is not a square number, then the terms of the continued fraction of 
satisfy
 | (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 
remaining after a given convergent the "tail," it must be true that the relationship between the number 
and terms in its tail is of the form
 | (3) |
which can only lead to a quadratic equation.
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.
An upper bound for the length of the period is roughly 
. The least positive 
s such that the continued fraction of 
has period 
, 2, ... are 2, 3, 41, 7, 13, 19, 58, 31, 106, ... (OEIS A013646). The first few values of 
such that the continued fraction of 
has period 
are summarized below for small 
.
 | OEIS |  |
| 1 | A002522 | 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, ... |
| 2 | A013642 | 3, 6, 8, 11, 12, 15, 18, 20, 24, 27, ... |
| 3 | A013643 | 41, 130, 269, 370, 458, ... |
| 4 | A013644 | 7, 14, 23, 28, 32, 33, 34, 47, 55, 60, ... |
| 5 | A010337 | 13, 29, 53, 74, 85, 89, 125, 173, 185, 218, ... |
| 6 | A020347 | 19, 21, 22, 45, 52, 54, 57, 59, 70, 77, ... |
| 7 | A010338 | 58, 73, 202, 250, 274, 314, 349, 425, ... |
| 8 | A020348 | 31, 44, 69, 71, 91, 92, 108, 135, 153, 158, ... |
| 9 | A010339 | 106, 113, 137, 149, 265, 389, 493, ... |
| 10 | A020349 | 43, 67, 86, 93, 115, 116, 118, 129, 154, 159, ... |
The values of 
at which the period of the continued fraction of 
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
(Wall 1948, pp. 39 and 83).
The first follows from
Therefore,
 | (13) |
so plugging (13) into (12) gives
 | (14) |
Expanding
 | (15) |
and solving using the quadratic formula gives
 | (16) |
The analog of this treatment in the general case gives
 | (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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