A lower bound for linear forms in values of a continued fraction

Every algebraic number except zero and unity (e), when raised to a power which is an irrational algebraic number gives a transcendental number.This was conjectured by Hilbert, and proposed as a problem in 1900.fIt was only recently that Gelfond,{ and independently at about the same time, Schneider, succeeded in proving this theorem.G. Ricci || has shown that the transcendentality persists if the algebraic numbers are replaced by products of algebraic numbers and a restricted type of Liouville number.The question we consider here is the more general one where the algebraic numbers are replaced by numbers capable of approximation by sequences of algebraic numbers in an appropriate manner.These form a restricted type of certain generalized Liouville transcendental numbers very closely related to those treated by Ore.^fThey bear a relation to the numbers of an algebraic field similar to that of the ordinary Liouville numbers to the rational numbers.Our principal result is that if X and Y are each suitably approximable by sequences of algebraic numbers, then XY and log X/log Y are transcendental.Some numerical examples are given in 7.Though the transcendental numbers found by us have the power of the continuum, they form a set of zero measure as we show.My interest in this subject was stimulated by a report on the Hilbert-Gelfond-Schneider theorem which I gave to the seminar of Professor H. Weyl at Princeton, and in particular by a conjecture of Professor J. von Neumann that natural generalizations of these results would involve approximations by algebraic numbers, rather than by rational numbers.I also wish to acknowledge my appreciation of the opportunity to work without distraction on this field, made possible by the courtesy of the Massachusetts Institute of Technology and the Institute for Advanced Study.

Every algebraic number except zero and unity (e°), when raised to a power which is an irrational algebraic number gives a transcendental number.This was conjectured by Hilbert, and proposed as a problem in 1900.fIt was only recently that Gelfond,{ and independently at about the same time, Schneider, § succeeded in proving this theorem.G. Ricci || has shown that the transcendentality persists if the algebraic numbers are replaced by products of algebraic numbers and a restricted type of Liouville number.The question we consider here is the more general one where the algebraic numbers are replaced by numbers capable of approximation by sequences of algebraic numbers in an appropriate manner.These form a restricted type of certain generalized Liouville transcendental numbers very closely related to those treated by Ore.^fThey bear a relation to the numbers of an algebraic field similar to that of the ordinary Liouville numbers to the rational numbers.Our principal result is that if X and Y are each suitably approximable by sequences of algebraic numbers, then XY and log X/log Y are transcendental.Some numerical examples are given in §7.Though the transcendental numbers found by us have the power of the continuum, they form a set of zero measure as we show.My interest in this subject was stimulated by a report on the Hilbert-Gelfond-Schneider theorem which I gave to the seminar of Professor H. Weyl at Princeton, and in particular by a conjecture of Professor J. von Neumann that natural generalizations of these results would involve approximations by algebraic numbers, rather than by rational numbers.I also wish to acknowledge my appreciation of the opportunity to work without distraction on this field, made possible by the courtesy of the Massachusetts Institute of Technology and the Institute for Advanced Study.

The obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

By using Pade approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are obtained. This estimate depends on all the coefficients of the linear form.

Lower and upper bounds that differ from each other only by a constant factor are obtained for linear forms in values of the function , and its successive derivatives at the point under the condition that , and are integers in some imaginary quadratic field.Bibliography: 9 titles.

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction ???????? are concentrated about fractions − ????????−2 ????????−1 is either in the interval of radius ???? (︁ 1 ????2 ????−1 )︁ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number ????0 = ????0(????), fair recurrent formula for incomplete private ???????? expansions of real algebraic irrationals ????, Express ???????? using the values of the minimal polynomial ????????−1(????) for residual fractions ????????−1 and its derivative at the point ????????−1. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of ???? in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − ????????−2 ????????−1 . It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number ???? and its limit points.

In this paper we first give the value of a periodic continued fraction which was recorded incorrectly by Ramanujan on page 341 of his lost notebook. Next, we describe several pairs of equivalent continued fractions in which one is the odd part of the other. One of the results is for the Rogers-Ramanujan continued fraction which was recently proved by Berndt and Yee. Finally, using the Bauer-Muir transformation we prove the equivalence of two continued fractions. One was recorded on page 44 in Ramanujan’s lost notebook, and the other is found in the unorganized pages at the end of Ramanujan’s second notebook.

The connections between the Golden Ratio namely (1 + √5)/2, a simple continued fraction, and Fibonacci and Lucas numbers, are familiar. The Fibonacci and Lucas numbers have many fascinating properties. We now point out that the square root of the Golden Ratio is the real part of a simple periodic continued fraction but using (complex) Gaussian integers a + ib instead of the natural integers. This fact provokes a definition and a study of complex Fibonacci and Lucas numbers, and the study again turns out to have a rich theoretic structure. A fuller account will appear in The Fibonacci Quarterly.

Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w n * (ξ) defined by Mahler and Koksma. We calculate their six values when n=2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction ξ by quadratic surds.

Ramanujan recorded many beautiful continued fractions in his notebooks. In this paper, we derive several indentities involving the Ramanujan continued fraction c(q), including relations between c(q) and c(qn). We also obtain explicit evaluations of for various positive integers n.

Let K = Q ( θ ) \mathcal {K} = \mathcal {Q}(\theta ) be the algebraic number field formed by adjoining θ \theta to the rationals Q \mathcal {Q} . Let R and h be, respectively, the regulator and class number of K \mathcal {K} . Shanks has described a method of evaluating R for Q ( D ) \mathcal {Q}(\sqrt D ) , where D is a positive integer. His technique improved the speed of the usual continued fraction algorithm for finding R by allowing one to proceed almost directly from the n th to the m th step, where m is approximately 2 n , in the continued fraction expansion of D \sqrt D . This paper shows how Shanks’ idea can be extended to the Voronoi algorithm, which is used to find R in cubic fields of negative discriminant. It also discusses at length an algorithm for finding R and h for pure cubic fields Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D}) , D an integer. Under a certain generalized Riemann Hypothesis the ideas developed here will provide a new method which will find R and h in O ( D 2 / 5 + ε ) O({D^{2/5 + \varepsilon }}) operations. When h is small, this is an improvement over the O ( D / h ) O(D/h) operations required by Voronoi’s algorithm to find R . For example, with D = 200171999 D = 200171999 , it required only 5 minutes for an AMDAHL 470/ V 7 computer to find that R = 518594546.969083 R = 518594546.969083 and h = 1 h = 1 . This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm.

If F F is a set of two or more positive integers, E ( F ) E(F) denotes the set of irrational real numbers x ∈ ( 0 , 1 ) x\in (0,1) such that x x can be expressed as a continued fraction which only uses the integers in F F . It is of interest to determine dim ⁡ ( E ( F ) ) \dim (E(F)) , the Hausdorff dimension of F F . Methods of R. S. Falk and R. D. Nussbaum (see references in this paper) give rigorous high order upper and lower bounds for dim ⁡ ( E ( F ) ) \dim (E(F)) . Here, we give some elementary “comparison theorems,” which sometimes allow one to prove that dim ⁡ ( E ( F 1 ) ) > dim ⁡ ( E ( F 2 ) ) \dim (E(F_1))>\dim (E(F_2)) without actually computing dim ⁡ ( E ( F 1 ) ) \dim (E(F_1)) or dim ⁡ ( E ( F 2 ) ) \dim (E(F_2)) .

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

When a particle dimension is comparable to the laser light wavelength, scattering calculations require the use of Mie or similar infinite series. The terms in the series may be an infinite set of complex determinants. Despite the power of modern computers these equations are difficult to compute with, because of the lack of analytical criteria for terminating the computations. Commonly, the series computations are stopped when the value of the series is changed by an amount less than a predetermined factor as a result of the addition of the computed term. This procedure has led to large errors. These series are easily transformed into the continued fraction form of the Fsad6 approximates. Complex continued fractions are more stable, computationally, than infinite series and frequently converge more rapidly. Further, an anlytical test exists for determining the circular domain within which the value of the continued fraction must lie. This permits an accurate and precise delineation of the maximum residual error. There is a disadvantage in the use of continued fractions. If the order of the creates convergent is not great enough for the desired accuracy, the computation must be repeated. This is not a serious handicap in actual use.

1. The Natural Numbers. The Basics. The Fundamental Theorem of Arithmetic. Searching for Primes. Number Fascinations. 2. The Integers. Diophantine Equations. Congruence Arithmetic. Pell and Pythagoras. Factoring Large Numbers. 3. The Rational Numbers. Rational Numbers as Decimals. Decimals as Rational Numbers. Continued Fractions. Solving Equations on the Rational Plane 4. The Real Numbers. Algebraic Representations. Geometric Representations. Analytic Representations. Searching for Transcendental Numbers. 5. Mathematical Projects. Rings of Factors. Sums of Consecutive Numbers. Measuring Abundance. Inside the Fibonacci Numbers. Pictures at an Iteration. Eenie Meenie Miney Mo. Factoring with the Pollard ...r Method. Charting the Integral Universe. Triangles on the Integral Lattice. The Gaussian Integers. Writing Fractions the Egyptian Way. Building Polygons with Dots. The Decimal Universe of Fractions, I. The Decimal Universe of Fractions, II. The Making of a Star. Making Your Own Real Numbers. Building 1 the Egyptian Way. Continued Fraction Expansions of x1/2 N. A Special Kind of Triangle. Polygon Numbers. Continued Fraction Expansions.