Related papers: On the Connection Between Irrationality Measures a…
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…
Recently Raayoni et al. announced various conjectures on continued fractions of fundamental constants automatically generated with machine learning techniques. In this paper we prove some of their stated conjectures for Euler number $e$ and…
The idea to use classical hypergeometric series and, in particular, well-poised hypergeometric series in diophantine problems of the values of the polylogarithms has led to several novelties in number theory and neighbouring areas of…
In this paper, Riemann's Zeta function with odd positive integer argument is represented as an infinite summation of integer powers of $\pi$ with rational coefficients. Specific values for Apery's Constant and Catalan's Constant are then…
Roger Apery's seminal method for proving irrationality is "turned on its head" and taught to computers, enabling a one second redux of the original proof of zeta(3), and many new irrationality proofs of many new constants, alas, none of…
We discuss the use of matrices for providing sequences of rationals that approximate algebraic irrationalities. In particular, we study the regular representation of algebraic extensions, proving that ratios between two entries of the…
An Engel series is a sum of reciprocals of a non-decreasing sequence $(x_n)$ of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence…
Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and…
In the proof of the irrationality of $\zeta(3)$ and $\zeta(2)$, Ap\'ery defined two integer sequences through $3$-term recurrences, which are known as the famous Ap\'ery numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced…
Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for…
For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…
Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still…
The Ramanujan Machine project predicts new continued fraction representations of numbers expressed by important mathematical constants. Generally, the value of a continued fraction is found by reducing it to a second order linear difference…
Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that 'tweaked' the sum of the $s^{th}$ powers of the $n$-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions,…
This paper presents a complete formal verification of a proof that the evaluation of the Riemann zeta function at 3 is irrational, using the Coq proof assistant. This result was first presented by Ap\'ery in 1978, and the proof we have…
We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic…
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…
In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_m\alpha - \frac{a_m}{c_m}\,\Big) \,,\] where $\alpha$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m…
For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the…
Let $k\geq 1$ be a small fixed integer. The rational approximations $\left |p/q-\pi^{k} \right |>1/q^{\mu(\pi^k)}$ of the irrational number $\pi^{k}$ are bounded away from zero. A general result for the irrationality exponent $\mu(\pi^k)$…