Pdf in this paper, we give a sufficient condition on two padic continued fractions. Continued fractions in local fields and nested automorphisms. Volume 101, number 3, november 1987 padic continued fractions of liouville type vichian laohakosol and patchara ubolsri communicated by larry j. The purpose of this paper is to extend this study of the metric theory of continued fractions to the setting of the p adic field for the rational prime p. Since the rational numbers are a subset of the adics, every rational number has a unique adic continued fraction which can be shown to be finite. And that means that we get collisions between fractions and negative numbers. Explicit methods for solving diophantine equations. Transcendental p adic continued fractions springerlink.
Next, we shall study the following set defined by the growth rate of a n x. P adic continued fractions request pdf researchgate. The padic numbers there are quite a few reasons to be interested in the padic numbers q p. As a consequence, any eld that can be endowed with an archimedean absolute value is isomorphic to a sub eld of c. The latter in turn constitute an extension of the eld of rational numbers, analogous to the completion of the rationals by the real numbers with respect to the standard ordinary metric.
In section 5 by using the open source software sage with p adic continued fractions and gauss reduction algorithm we show some numerical results, investigating and comparing these two algorithms. Note that in this field of fractions, every noninteger padic number can be uniquely written as p. A well known theorem of lagrange states that the simple continued fraction expansion of a real number is periodic if. A more thorough account of p adic continued fraction algorithms can be found in the 2007 ph. Dirichlet also added to the general theory, as have numerous contributors to the applications of the. The continued fraction expansion on the padic numbers ergodic properties of the padic continued fraction map means of padic continued fraction maps entropy of the p adic continued fraction map isomorphism of dynamical systems ornsteins theorem non archemedean elds examples continued fraction maps on the eld of formal power series. Is it useful and does it have relations to best rational approximation in the p adic sense. Before we can ask why q p is important, we must ask why these new metrics one for. Hensel though they are foreshadowed in the work of his predecessor e. The p adic integers form a subset of the set of all p adic numbers.
Dec 12, 2012 multiplication and division get even stranger in padic. The intention of this paper is to present some interesting results concerning su. On the metric theory of padic continued fractions sciencedirect. But the theory is somewhat unsatisfying, imo, as the set of p adic numbers with finite continued fraction does not form a ring not even an additive group, and is. In the analytic case there is the p fraction and the associated padeapproximants. Ruban in certain metric properties of the p adic numbers. Are there any explicitly known algebraic numbers with unbounded continued fraction coefficients. Building on this work, we establish similar results in the case of development in padic continued fractions.
This demonstration computes the adic continued fractions for all rational numbers of the form. This means that natural numbers are exactly the same thing as padic integer only a. This website uses cookies to ensure you get the best experience on our website. Continued fractions 1992 pdf niv, reference bible, large print, hardcover. A padic analogue of a recent algebraic independence criterion of adams is proved.
On the other hand, there is a much larger variety of elds with a nonarchimedean absolute value. For p2, we give a combinatorial characterization of rational numbers that have terminating expansions. Hensel, and this invention led to substantial developments in number theory, where padic numbers are now as natural as ordinary real numbers. Is there an analogue of the simple continued fraction in p adic number fields. Fractions and padic numbers real numbers and limits math. Explicit examples of padic numbers with prescribed. Continued fractions, closely related to irrational numbers and due to cataldi, 16, received attention at the hands of euler, and at the opening of the 19th century were brought into prominence through the writings of josephlouis lagrange. See,, for more dimensional results on the sum of digits of continued fractions of real numbers. Padic continued fractions, the ramanujan journal 10. They are useful for solving diophantine equations, using tools like hensels lemma and the hasse principle, which we wont say anything about.
May 26, 2011 we study schneiders padic continued fraction algorithms. This is a wellwritten introduction to the world of p adic numbers. In, several algorithms which generalize the continued fractions over the padic numbers and a complete bibliography of the argument are shown. However, no algorithm has been found such that it always produces a periodic representation for every square root in the field of padic numbers. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. At each stage there is a unique solution, so after xing the initial choice. We establish a new transcendence criterion of padic continued fractions which are called ruban continued fractions. We see that this algorithm is derived from a generalized padic euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers. Mahler in zur approximation padischer irrationalzahlen and a. The adic continued fraction of a adic number is similar to the usual simple continued fraction in the reals with the requirement that. By this result, we give explicit transcendental ruban continued fractions with bounded padic absolute value of partial quotients.
I we will see, among others, the p adic analogue of classical continued fractions as a particular case of nested automorphism and the approximation lattices. The main goal of this paper is to consider the analogous problem for p adic numbers here, and throughout the paper, pdenotes a prime number, that is, to construct explicit examples of p adic numbers with prescribed irrationality exponent and only digits 0 and 1 in their hensel expansion. The euclidean algorithm can be used to find simple continued fractions of a rational number. Padic numbers 5 ostrowski proved that any eld complete with respect to an archimedean absolute value is isomorphic to r or c. Browkin 1 studied a construction of certain type of continued fractions which. This video is an exploratory video in which we loosely introduce an interesting variant on repeating decimals. Madore revised 7th december 2000 in all that follows, pwill stand for a prime number. Building on this work, we establish similar results in the case of development in p adic continued fractions. The padic integers will form a ring z p, which contains z. A course on number theory pdf 9p this note explains the following topics.
A padic continued fraction is a fraction of the form. Apr 29, 2012 this video is an exploratory video in which we loosely introduce an interesting variant on repeating decimals. In addition we show irrationality of some other padic lseries values, and values of the padic hurwitz zetafunction. N, z, q, rand care the sets of respectively the natural numbers i. Is there an analogue of the simple continued fraction in padic number fields. I we will also generalize in these cases, when it is possible, the. Padic numbers article about padic numbers by the free. It is then applied to construct liouville type padic continued. On the digits of schneiders padic continued fractions. For p2, we give a combinatorial characterization of rational numbers that have. Apr 12, 2014 the first part develops the fraction from a simple equation or statement with a single unknown variable and demonstrates the recursive, iterative procedure.
Wikipedia is longer, has more examples, and shows both the analytical and algebraic views. Applications of finite continued fractions mathoverflow. In section 5 by using the open source software sage with padic continued fractions and gauss reduction algorithm we show some numerical results, investigating and comparing these two algorithms. My interests are in computational number theory, padic cohomology, elliptic curves and modular forms. On the metric theory of adic continued fractions request pdf. Padic continued fractions padic continued fractions hirsh, jordan. In mathematics, the padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. Numbers and functions as continued fractions numericana.
They are basic examples for things like valuations and pro nite groups. Is it useful and does it have relations to best rational approximation in the padic sense. However, in section 9 we prove a new result, namely a padic analogue. May 26, 2011 padic continued fractions padic continued fractions hirsh, jordan. The ring of padic integers has no zero divisors, so we can take the field of fractions to get the field q p of padic numbers.
Ruban in certain metric properties of the padic numbers. We all know that p 2 is irrational, so that 2 is not a square in the. The subspace theorem is a multidimensional generalization of the theorem of. The pade approximants are best rational approximants in the x adic metric.
I can prove that under this function each rational padic number maps to a rational padic number, and each. We study schneiders padic continued fraction algorithms. Continued fractions for which the sequence of partial quotients is ultimately periodic are called periodic continued fractions and they correspond to quadratic irrationals also called algebraic numbers of degree 2, these are irrational roots of polynomials of degree 2 with integral coefficients. The pade approximants are best rational approximants in the xadic metric.
Continued fractions over the padic numbers have been also studied recently in several works. This lecture note is an elementary introduction to number theory with no algebraic prerequisites. But the theory is somewhat unsatisfying, imo, as the set of padic numbers with finite continued fraction does not form a ring not even an additive group, and is. An introduction to algebraic number theory download book. Continued fractions in p adic numbers by michitaka kojima abstract the theory of continued fractions of irrational numbers is developed in the eld r of real numbers, and there are many well known results on this subject. Irrationality, padic lseries, continued fraction ams 2000 classi. It seems that hensels main motivation was the analogy between the ring of integers z, together with its. P adic numbers 5 ostrowski proved that any eld complete with respect to an archimedean absolute value is isomorphic to r or c. Algebraic numbers, finite continued fractions, infinite continued fractions, periodic continued fractions, lagrange and pell, eulers totient function, quadratic residues and nonresidues, sums of squares and quadratic forms. A crucial difference is that the reals form an archimedean field, while the p padic numbers form a nonarchimedean field. You did not limit the context of continued fractions to numbers. In section 3 we introduce approximation lattices of padic numbers and the shortest vectors. The continued fraction expansion on the padic numbers ergodic properties of the padic continued fraction map means of padic continued fraction maps entropy of the p adic continued fraction map isomorphism of dynamical systems ornsteins theorem non archemedean elds examples continued fraction maps on the eld of formal power series classi.
Continued fractions in padic numbers 241 so we need to modify the expression 1. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions. The reader is led into the rich structure of the fields qp and cp in a beautiful balance between analytic and algebraic aspects. In section 4 we study the quasiperiodic dynamical system. In addition we show irrationality of some other p adic lseries values, and values of the p adic hurwitz zetafunction. The main goal of this paper is to consider the analogous problem for padic numbers here, and throughout the paper, pdenotes a prime number, that is, to construct explicit examples of padic numbers with prescribed irrationality exponent and only digits 0 and 1 in their hensel expansion. The aim of this talk is to give a survey on padic numbers and padic analysis. Continued fractions for which the sequence of partial quotients is ultimately periodic are called periodic continued fractions and they correspond to quadratic irrationals also called algebraic numbers of degree 2, these are irrational roots of polynomials of degree 2 with integral. In this paper we present an approach using classical continued fractions discovered by stieltjes. A simple continued fraction is a fraction of the form. Is the analogue of the simple continued fraction in padic. For p2, we give a combinatorial characterization of rational numbers that. In section 3 we introduce approximation lattices of p adic numbers and the shortest vectors.
Because we cant have an infinite number of digits to the right of the decimal, padic ends up representing fractions using infinite numbers of digits on the right of the decimal. Purely periodic expansions as a warmup, lets describe padic numbers with. Solution of pell equation over field of padic numbers. The purpose of this paper is to describe the padic numbers, but so far we have only established what the padic metric is. Before we can ask why q p is important, we must ask why these new metrics one for each prime p are important. Then continued fractions can be used whenever you have a euclidian division, preferably when there is a natural choice of quotient remainder, so that it is done in a unique way. Rational numbers correspond to finite continued fractions. We use n, z, q to denote the natural numbers, the integers and the rational numbers respectively.
It contrasts with the result in the case of schneiders padic continued fractions, since dim h. Finally, we prove an analogue of khinchins theorem. In this paper, we discuss some of the corresponding results for continued fractions of irrational numbers in the p adic. Meanwhile the theory of padic numbers has a plenty of applications and impacts in various mathematical. I am a university lecturer in the mathematical institute, with the title of professor of mathematics, and the tutorial fellow in pure mathematics at hertford college. Mahler in zur approximation p adischer irrationalzahlen and a. An analogue of the regular continued fraction expansion for the adic numbers for prime was given by t. For a fixed prime number p, a fraction of the form a p k, where a is a p adic integer and k is a nonnegative integer.
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