Newton's Formula and the Continued Fraction Expansion of √d

We prove a conjecture of Drake and Kim on a continued fraction. Comment: 6 pages, 2 figures

Humanistic Mathematics Network Journal, 1999

Proceedings of the Edinburgh …, 1991

2013

Abstract. If the equation of the title has an integer solution with k ≥ 2, then m> 109.3·106. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark m> 10107. Here we achieve m> 10109 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N, with N an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application. 1.

Acta Mathematica, 2005

Indagationes Mathematicae (Proceedings), 1983

The following conjecture of H.W. Lenstra is proved. Denote by pn/qn, n = 1,2,. .. the sequence of continued fraction convergents of the irrational number x and define 6,(x): =qnjqnx-p,I. Then for every z, 01 ZI 1, one has for almost all x 2 r 3 lim !. #{j; j~~,Bi(x)~~} = log 2 oszs+ no) n & (-z+log 2z+ l), +:zs1 Similar results are proved for other functions connected with the regular continued fraction expansion, such as the quotient of as well as for other type of expansions, such as the nearest integer and singular continued fractions. The main tool is the natural extension of the operator x-(1/x)-[(l/.x)], recently studied by Hitoshi Nakada.

Rocky Mountain Journal of Mathematics, 2003

Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n) which converge faster. The estimation of the improvement and bounds for the error |f − S n (w n)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.

arXiv (Cornell University), 2021

A Trott number is a number x ∈ (0, 1) whose continued fraction expansion is equal to its base b expansion for a given base b, in the following sense: If x = [0; a 1 , a 2 ,. .. ], then x = (0.â 1â2. . .) b , whereâ i is the string of digits resulting from writing a i in base b. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set T b of Trott numbers is a complete G δ set. We prove moreover that the union T := b≥2 T b is nowhere dense and has Hausdorff dimension less than one. Finally, we give several sufficient conditions on bases b and b ′ such that T b ∩ T b ′ = ∅, and conjecture that this is the case for all b = b ′. This question has connections with some deep theorems in Diophantine approximation.

Acta Arithmetica, 2007

In this paper we show how to construct several infinite families of polynomials D(x, k), such that p D(x, k) has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter k. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.