Related papers: Another Proof of $e^{x/y}$ being irrational
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.
We prove that the formula giving the exact value of the irrationality exponent of regular continued fractions remains valid for semi-regular continued fractions satisfyiong certain conditions.
We apply a paraconsistent logic to reason about fractions.
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution.
In this paper we review a general proof for the irrationality property of numbers which take a certain form of infinite sums.
An application of (iterated) Bauer-Muir acceleration can give an Ap\'ery-like continued fraction for $\pi$ with irrational coefficients, and much faster convergence. It can be considered a generalized continued fraction with the same matrix…
A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…
Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we…
We illustrate the power of Experimental Mathematics and Symbolic Computation to suggest irrationality proofs of natural constants, and the determination of their irrationality measures. Sometimes such proofs can be fully automated, but…
We provide an alternative proof that $\sqrt{2}$ is irrational that does not begin with the assumption that $\sqrt{2}$ is in fact rational.
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not…
We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Ap\'ery in his proof of the irrationality of $\zeta(3)$. We show in particular that this can be applied to a…
We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q >…
We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary…
Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then…
We prove that a real number a greater than or equal to 2 is the irrationality exponent of some computable real number if and only if a is the upper limit of a computable sequence of rational numbers. Thus, there are computable real numbers…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.