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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.
We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the evaluations to be expressed in terms of zeta values or more general poly-logarithmic terms. The machinery developed is then… (More)
We provide a fairly simple and straightforward argument yielding all substitution invariant Beatty sequences.
In his 'Memoir on Elliptic Divisibility Sequences', Morgan Ward's definition of the said sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial such sequences. Even then, Ward's proof of coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß… (More)
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)
In memory of Alf van der Poorten ABSTRACT. In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat's little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests… (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)
Using WZ pairs we present accelerated series for computing (3) AMS Subject Classiication: Primary 05A Alf van der Poorten P] gave a delightful account of Ap ery's proof A] of the irrationality of (3). Using WZ forms, that came from WZ1], Doron Zeilberger Z] embedded it in a conceptual framework. We recall Z] that a discrete function A(n,k) is called… (More)
Arithmetic properties of integer sequences counting periodic points are studied, and applied to the case of linear recurrence sequences, Bernoulli numerators, and Bernoulli denominators .
We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series c h X −h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple… (More)