We exhibit uncountably many binary decimals together with their explicit continued fraction expansions. These expansions require only the partial quotients 1 or 2. The pattern of valleys and ridges in a sheet of paper repeatedly folded in half plays a critical rôle in our construction.
Let p be a prime congruent to 1 modulo 4, and let t, u be rational integers such that (t + u √ p)/2 is the fundamental unit of the real quadratic field Q(√ p). The Ankeny-Artin-Chowla conjecture (AAC conjecture) asserts that p will not divide u. This is equivalent to the assertion that p will not divide B (p−1)/2 , where Bn denotes the nth Bernoulli number.… (More)
In this paper, we discuss issues related to the efficient implementation of Shanks' NUCOMP algorithm for computing the reduced composite of two binary quadratic forms. In particular, we describe how efficient versions of NUCOMP can be implemented for computations in imaginary quadratic number and function fields, as well as computations in the… (More)
We provide a fairly simple and straightforward argument yielding all substitution invariant Beatty sequences.
Though my travels took a long t˘ ıme, I hope Paulo will think it is f˘ ıne For my remarks to be short; 'Cause the point is the thought That I write this for P. Ribenboim
We detail the continued fraction expansion of the square root of the general monic quartic polynomial. We note that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. The paper includes a detailed 'reminder exposition' on continued fractions of quadratic irrationals in… (More)
It is well known that one can obtain explicit continued fraction expansions of e z for various interesting values of z ; but the details of appropriate constructions are not widely known. We provide a reminder of those methods and do that in a way that allows us to mention a number of techniques generally useful in dealing with continued fractions.… (More)
There is a class of quadratic number fields for which it is possible to find an explicit continued fraction expansion of a generator and hence an explicit formula for the fundamental unit. One there-with displays a family of quadratic fields with relatively large regulator. The formula for the fundamental unit seems far simpler than the continued fraction… (More)
We note that the continued fraction expansion of a lacunary formal power series is a folded continued fraction with monomial partial quotients, and with the property that its convergents have denominators that are the sums of distinct monomials, that is, they are polynomials with coefficients 0, 1, and −1 only. Our results generalise, simplify and refine… (More)