Related papers: A determinantal approach to irrationality

Given a rational number $r$ such that $2r$ is not an integer, we prove that $\tan^2(r\pi)$ is irrational unless it is equal to $0$, $1$, $3$ or $\frac{1}{3}$, using only basic trigonometry and the Rational Root Theorem. Moreover, we deduce…

Two types of explanations have been receiving increased attention in the literature when analyzing the decisions made by classifiers. The first type explains why a decision was made and is known as a sufficient reason for the decision, also…

Continued fractions are used to give an alternate proof of $e^{x/y}$ is irrational.

Qualitative and quantitative approaches to reasoning about uncertainty can lead to different logical systems for formalizing such reasoning, even when the language for expressing uncertainty is the same. In the case of reasoning about…

Let $p$ be a prime number and $\xi$ an irrational $p$-adic number. Its multiplicative irrationality exponent ${{\mu^{\times}}} (\xi)$ is the supremum of the real numbers ${{\mu^{\times}}}$ for which the inequality $$ |b \xi - a|_{p} \leq |…

Let X be a complex, rationally connected, projective manifold. We show that X admits a modification X' that contains a quasi-line, ie a smooth rational curve whose normal bundle is a direct sum of copies of O_{P^1}(1). For manifolds…

In this paper, we establish new irrationality criteria for certain sparse power series. As applications of these criteria, we generalize a result of Erd\H{o}s and obtain several irrationality results for various infinite series involving…

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a…

We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an…

In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing…

Solutions of Pell's equation and hypergeometric series identities are used to study series expansions for $\sqrt{p}$ where $p$ are arbitrary prime numbers. Numerous fast convergent series expansions for this family of irrational numbers are…

In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…

The degree of irrationality $irr(X)$ of a $n$-dimensional complex projective variety $X$ is the least degree of a dominant rational map $X\dashrightarrow \mathbb{P}^n$. It is a well-known fact that given a product $X\times \mathbb{P}^m$ or…

The main purpose of this article is to get a handle on determining how far a non-rational singularity is from being rational, or in other words, introduce a measure of the failure of a singularity being rational.

From Sturmian and Christoffel words we derive a strictly increasing function $\Delta:[0,\infty)\to\mathbb{R}$. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous.…

Based upon the axiom of choice it is proved that the cardinality of the rational numbers is not less than the cardinality of the irrational numbers. This contradicts a main result of transfinite set theory and shows that the axiom of choice…

Let X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number…

Propositional inquisitive logic is the limit of its $n$-bounded approximations. In the predicate setting, however, this does not hold anymore, as discovered by Ciardelli and Grilletti, who also found complete axiomatizations of $n$-bounded…

The L\'evy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic…

In this paper, we simplify and improve the constant, $c$, that appears in effective irrationality measures, $|(a/b)^{m/n}-p/q|>c|q|^{-(\kappa+1)}$, obtained from the hypergeometric method for $a/b$ near $1$. The dependence of $c$ on $|a|$…