There must be a more convenient way to describe a continued fraction. Period of the continued fraction of v n marius beceanu february 5, 2003 abstract this paper seeks to recapitulate the known facts about the length of the period of the continued fraction expansion of v n as a function of n and to make a few possibly original contributions. Any periodic continued fraction represents a root of a quadratic equation with integer coefficients. The periodic properties of continued fractions are discussed in chapter 4. Periodic continued fractions millersville university of. Unlike decimal fractions, which depend on the choice of base ten, continued fractions are free of arti cial choices. The main part of the chapter develops a proof of lagranges theorem that the continued. A continued fraction is a fraction in which the numerators and denominators may contain continued fractions. Periodic continued fractions, quadratic irrationalities mit. Moreover, one part of the proof will require the construction of an algorithm for computing the continued fraction expansion for a quadratic. 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. Random walks and periodic continued fractions volume 17 issue 1 wolfgang woess.
Sergey khrushchev and michael tyaglov international school of economics kazakhbritish technical university almaty, kazakhstan and shnaghai university, china 09112012 osu sergey khrushchev and michael tyaglov. Suppose we start with a rational number, then euclids algorithm terminates in nitely. If the sequences of integers appearing in the continued fraction expansion contain all finite sequences of integers, then the geodesic is dense. Periodic and limit periodic continued fractions springerlink. Periodic simple continued fraction quadratic irrational like. Robertson, on purely periodic nearest square continued fractions, journal of combinatorics and number theory, 2, issue 3 2010 239244. Nelsen 2018 periodic continued fractions via a proof without words mathematics magazine 91. In section 5 we write explicitly quartic polynomials with square leading coefficients such that the square root has a periodic continued fraction expansion over a quadratic extension of q. Continued fractions world scientific publishing company.
Proceedings of the steklov institute of mathematics, vol. Every rational number has a simple continued fraction expansion which is finite and every finite simple. Continued fractions and pells equation seung hyun yang abstract. Period of the continued fraction of v n princeton math. Periodic continued fractions and elliptic curves over quadratic. Another important theorem, due to liouville, allows us to construct transcendental numbers by simply taking continued fractions whose terms grow fast enough kh61, theorem 27. Lecture 1 we describe the farey tessellation fand give a very quick introduction to the basic facts we need from hyperbolic geometry, using the upper half plane model. Continued fractions and periodic jacobi matrices stephen a. All the numerators are 1s, so all we need to do is list the denominators. Continued fractions are visually striking as they slide down to the right, but writing them as fractions takes a lot of ink and a lot of space. In this reu paper, i will use some important characteristics of continued fractions to give the complete set of solutions to pells equation. Periodic continued fractions jordan schettler abstract. Periodic continued fractions introduction purely periodic continued fractions quadratic irrationals reduced quadratic irrationals converse of theorem 4.
We can determine when they converge, when they diverge, their values if they converge and the asymptotic behavior of their tail sequences. Lagrange used continued fractions to find the value of irrational roots. Continued fractions are a natural way of expressing irrational numbers. A quadratic irrational number x has a purely periodic continued fraction if and only. We prove an analog of lagranges theorem for continued fractions on the heisenberg group. I have established a result concerning the average period length. In either case, all integers in the sequence, other than the first, must be positive. Every rational number has a simple continued fraction expansion which is nite and every nite simple continued fraction expansion is a rational number. A periodic continued fraction is one which repeats for example, in general, a periodic continued fraction has the form if n is the length of the smallest repeating part, we say that the period is n.
While 1 looks into the option of integer numerator values, we wish to go further. Hardly any other idea has stimulated the mind so fruitfully in a certain sense, mathematical analysis is a symphony of the in nite. Continued fractions, fermat, euler, lagrange introduction. Proving that if x is a quadratic irrational, then continued fraction is periodic. The main part of the chapter develops a proof of lagranges theorem that the continued fraction expansion of every quadratic irrational is periodic after a. The continued fraction expansion of a real number is an alternative to the representation of such a. Random walks and periodic continued fractions advances in. He also proved that a real root of a quadratic irrational is a periodic continued fraction. Periodic continued fractions are quadratic irrationals fold unfold. Finally, because periodic continued fractions have infinitely many terms. The algorithm stops after finitely many steps if and only if x is rational. Brualdi abstract we describe an algorithm, based on a.
In this talk we introduce continued fractions, prove their basic properties and apply these properties to solve a practical problem. We also state without proof some further properties of continued fractions and provide a brief overview of some facts in this connection. The reader will find this chapter more challenging than the others, but the end results are rewarding. Available formats pdf please select a format to send. Continued fractions offer a means of concrete representation for arbitrary real numbers. Let be convergents of a nonsimple continued fraction. This, however, is not the case since p 3944 0,1,16,11,1,3,2,3,1,11,16,2. However, the same is not true for more general forms of continued fractions. Continued fractions, pells equation, and other applications jeremy booher continued fractions usually get shortchanged at promys, but they are interesting in their own right and useful in other areas of number theory. Proving that if xis a quadratic irrational, then continued fraction is periodic step 0. Ergodic theory of simple continued fractions robert hines september 3, 2015 1 simple continued fractions every irrational number x2rnq has a unique representation of the form. The main part of the chapter develops a proof of lagranges theorem that the continued fraction expansion of every quadratic irrational is periodic after a certain. Let e be an elliptic curve defined over a field k with char k.
Continued fractions have been studied for over two thousand years, with one of the first recorded studies being that of euclid around 300 bc in his book elements when he used them to find the greatest common divisor of two integers using what is known today as the euclidean algorithm. The nineteenth century can probably be described as the golden age of continued fractions. If the continued fraction expansion of p fx is periodic, then its period nlies in the set. Purely periodic and transcendental complex continued. First, i need to be able to write a quadratic irrational in a standard form. Periodic continued fractions, quadratic irrationalities, lecture 20. For general positive integers n, it reduces to a new iteration of an ndimensional simplex. Although we do not limit ourselves to this case in the course, it will be used. So we have another example of the unpredictability of continued fractions we mentioned above. Preface these are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. Apr 26, 2019 if the inline pdf is not rendering correctly. In nite continued fractions from time immemorial, the in nite has stirred mens emotions more than any other question. On the van vleck theorem for limitperiodic continued fractions of general form.
The primary result of this section is a theorem of lagrange which. Since then, continued fractions have shown up in a variety of other areas, including, but not limited to. Brualdi abstract we describe an algorithm, based on a continued fraction expansion, to reconstruct. We have the more general result presented in 7 that states. Richard carson school of physics and astronomy, university of st andrews, st. Continued fractions i continued fractions different way to represent real numbers. Pdf a few classes of periodic continued fractions researchgate. Continued fractions can also be used to calculate gear ratios, and were used for this purpose by the ancient greeks guy 1990. Periodic continued fractions, quadratic irrationalities.
The above expansion is called the simple continued fraction of x. If only the first few terms of a continued fraction are kept, the result is called a convergent. Random walks and periodic continued fractions advances. In a finite continued fraction or terminated continued fraction, the iterationrecursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. Basic definitions and facts a continued fraction is given by two sequences of numbers bnn. For example, if the continued fractions are periodic, then the geodesic will be periodic a closed loop as well. Displayed in their full laddered form, they look like. Pdf on jan 1, 1986, claude levesque and others published a few classes of periodic continued fractions find, read and cite all the. Periodic continued fractions are quadratic irrationals. Indeed, it is all a matter of iterations of linear fractional transformations, and we have closed expressions for their approximants s n w. The proof that quadratic irrationals give rise to periodic continued fractions will come out of an algorithm for computing the continued fraction for a quadratic irrational, which is useful in its own right.
For example, they given a way to write a prime congruent to 1 modulo 4 as a sum of two squares. Basic notations in general, a simple continued fraction is an. On the van vleck theorem for limit periodic continued fractions of general form. Then we will limit our consideration to continued fractions. We call the minimal such the period of the continued fraction. Thus patterns in the continued fraction expansions have a universal and deeper meaning. One traditional way to write a continued fraction is. The goals of this project are to have the reader explore some of the basic properties of continued fractions and prove that 2 r is a quadratic irrational i is equal to a periodic continued fraction. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and applications are included the. Besides new and simpler proofs for many of the standard topics, numerous numerical examples and. Apr 26, 2019 these socalled periodic continued fractions are precisely those that represent quadratic irrationalities.
We present a test for determining whether a real quadratic irrational has a purely periodic nearest square continued fraction expansion. The goals of this project are to have the reader explore some of the basic properties of continued fractions and prove that 2r is a quadratic irrational i is equal to a periodic. We must acknowledge that the set of numbers that have simply periodic, almost symmetrical continued fractions is much larger than the one mentioned above. This book presents the arithmetic and metrical theory of regular continued fractions and is intended to be a modern version of a. On continued fractions of the square root of prime numbers. Periodic continued fractions and elliptic curves over.
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