Proof Of Irrationality Research Articles

Mellin transforms for multiple Jacobi–Piñeiro polynomials and a [formula omitted]-analogue

This work treats the Mellin transform of multiple Jacobi–Piñeiro polynomials. This allows us to put a number of irrationality and Q -linear independence proofs into the framework of Hermite–Padé approximation. A similar approach is presented for the q -analogue: the q -Mellin transform of multiple little q -Jacobi polynomials and its applications in irrationality proofs.

Irrationality proof of certain Lambert series using little q-Jacobi polynomials

We apply the Padé technique to find rational approximations to h ± ( q 1 , q 2 ) = ∑ k = 1 ∞ q 1 k 1 ± q 2 k , 0 < q 1 , q 2 < 1 , q 1 ∈ Q , q 2 = 1 / p 2 , p 2 ∈ N ∖ { 1 } . A separate section is dedicated to the special case q i = q r i , r i ∈ N , q = 1 / p , p ∈ N ∖ { 1 } . In this construction we make use of little q -Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of h ± ( q 1 , q 2 ) and give an upper bound for the irrationality measure.

Decimal periods and their tables: A German research topic (1765–1801)

At the beginning of the 18th century, several mathematicians noted regularities in the decimal expansions of common fractions. Rules of thumb were set up, but it was only from 1760 onward that the first attempts to try to establish a coherent theory of periodic decimal fractions appeared. J.H. Lambert was the first to devote two essays to the topic, but his colleagues at the Berlin Academy, J. III Bernoulli and J.L. Lagrange, also spent time on the problem. Apart from the theoretical side of the question, the applications (factoring, irrationality proofs, and computational advantages), as well as the tabulation of decimal periods, aroused considerable interest, especially among Lambert's correspondents, C.F. Hindenburg and I. Wolfram. Finally, in 1797–1801, the young C.F. Gauss, informed of these developments, based the whole theory on firm number-theoretic foundations, thereby solving most of the open problems left by the mathematicians before him.

Irrationality proof of a q-extension of ζ(2) using little q-Jacobi polynomials

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 integer greater than one. These approximants are good enough to show the irrationality of $\zeta_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $\mu(\zeta_q(2))\leq 10\pi^2/(5\pi^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $\zeta(2)$}, Mat. Sb. \textbf{193} (2002), no. 8, 49--70).

The Peikovian Doctrine of the Arbitrary Assertion

Abstract The doctrine of the arbitrary assertion is a key part of Objectivist epistemology as elaborated by Leonard Peikoff. For Peikoff, assertions unsupported by evidence are neither true nor false; they have no context or place in the hierarchy of conceptual knowledge; they are meaningless and paralyze rational cognition; their production is proof of irrationality. A thorough examination of the doctrine reveals worrisomely unclear standards of evidence and a jumble of contradictory claims about which assertions are arbitrary, when they are arbitrary, and what ought to be done about them when they are. A wholesale rejection of the doctrine is recommended.

The Peikovian Doctrine of the Arbitrary Assertion

Abstract The doctrine of the arbitrary assertion is a key part of Objectivist epistemology as elaborated by Leonard Peikoff. For Peikoff, assertions unsupported by evidence are neither true nor false; they have no context or place in the hierarchy of conceptual knowledge; they are meaningless and paralyze rational cognition; their production is proof of irrationality. A thorough examination of the doctrine reveals worrisomely unclear standards of evidence and a jumble of contradictory claims about which assertions are arbitrary, when they are arbitrary, and what ought to be done about them when they are. A wholesale rejection of the doctrine is recommended.

Proofs of the irrationality of e

Proofs of the irrationality of <i>e</i>

A Collection of Numbers Whose Proof of Irrationality Is like That of the Number e

A Collection of Numbers Whose Proof of Irrationality Is like That of the Number e

Irrationality of Square Roots

Indeed, if or were a fraction with denominator q, then m +na would also be fraction with denominator q. Such a fraction is either zero or at least \/q in magnitude. Let us first note some previous proofs of irrationality based on this criterion. Arbi trarily small numbers m + na have been constructed using Euclid's Algorithm, starting with a and 1. To prove that one gets arbitrarily small nonzero numbers, one must show that the sequence of numbers produced by Euclid's algorithm does not terminate. For certain numbers a, this can be done by finding a pair of consecutive numbers whose ratio is the same as the ratio of a previous pair of consecutive numbers. The sequence of ratios of consecutive numbers is periodic from then on. Kaiman, Mena, and Shariari [1] give a geometric proof that this sequence is periodic for a = \?2. Geometric proofs must be tailored to each specific number and they are bound to get very complicated. For instance, one can show by computation that for a = \/43, the ratios repeat only after 10 steps; a geometric proof would therefore have to contain dozens of points and line segments. Using algebra, Joseph Louis Lagrange proved that Euclid's algorithm is periodic for all square roots. This result can be found in books discussing continued fractions. For other kinds of irrationals such as cube roots Euclid's algorithm is not periodic and the author does not know of a direct way of showing it will not terminate.

A proof of the irrationality of π and the rational powers of e

In this note we give a very short proof of irrationality of π, and a proof of the irrationality of the rational powers of e . The canonical proof of the irrationality of π is due to Lambert (1767), and the one fore to Euler (1737).

WELL-POISED HYPERGEOMETRIC TRANSFORMATIONS OF EULER-TYPE MULTIPLE INTEGRALS

Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannot be achieved by evident changes of variable. All this generalizes some classical results of Whipple and Bailey in analysis and, on the other hand, certain analytic constructions with known connection to irrationality proofs for values of Riemann’s zeta function at positive integers. 1. Motivation from number theory

Computerized deconstruction

The inequality (DERRIDA+TURING)>(DERRIDA)+(TURING) will be illustrated by computerized deconstruction of Roger Apéry's miraculous proofs of irrationality.

Random Generators and Normal Numbers

Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [Bailey and Crandall 01], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results are achieved: We establish b-normality for constants of the form Σ 1/(b mi c ni) for certain sequences (mi), (ni) of integers. This work unifies and extends previously known classes of explicit normals. We prove that for coprime b, c > 1 the constant α b,c = Σ n=c,c 2,c 3 , … 1/(nb n ) is b-normal, thus generalizing the Stoneham class of normals [Stoneham 73a]. Our approach also reproves b-normality for the Korobov class [Korobov 90] β b, c d , for which the summation index n above runs instead over powers cd , cd2 , cd3 , … with d > 1. Eventually we describe an uncountable class of explicit normals that succumb to the PRNG approach. Numbers of the α,β classes share with fundamental constants such as π, log 2 the property that isolated digits can be directly calculated, but for these new classes such computation tends to be surprisingly rapid. For example, we find that the googol-th (i.e., 10100 -th) binary bit of α2,3 is 0. We also present a collection of other results—such as digit-density results and irrationality proofs based on PRNG ideas—for various special numbers.

Similarities in Irrationality Proofs for π, ln2, ζ(2), and ζ(3)

Similarities in Irrationality Proofs for π, ln2, ζ(2), and ζ(3)

Similarities in Irrationality Proofs for π, ln2, ζ(2), and ζ(3)

Similarities in Irrationality Proofs for π, ln2, ζ(2), and ζ(3)

Little q-Legendre polynomials and irrationality of certain Lambert series

Certain q-analogs h p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős (J. Indiana Math. Soc. 12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory 37, 1991, 253–259; Proc. Cambridge Philos. Soc. 112, 1992, 141–146) used Pade approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln p (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apery's proof of irrationality of ζ(2) and ζ(3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Vaananen (Compositio Math. 91, 1994, 175–199) and recently also by Matala-aho and Vaananen (Bull. Australian Math. Soc. 58, 1998, 15–31) (for ln p (2)). In this paper we show how one can obtain rational approximants for h p(1) and ln p (2) (and many other similar quantities) by Pade approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Vaananen for h p(1) and a better upper bound as the one given by Matala-aho and Vaananen for ln p (2).

On Lambert's Proof of the Irrationality of &amp;#960

On Lambert's Proof of the Irrationality of &amp;#960

Math Bite: A Quick Counting Proof of the Irrationality of

Math Bite: A Quick Counting Proof of the Irrationality of

Rational approximations to the dilogarithm

The irrationality proof of the values of the dilogarithmic function L 2 ( z ) {L_2}(z) at rational points z = 1 / k z = 1/k for every integer k ∈ ( − ∞ , − 5 ] ∪ [ 7 , ∞ ) k \in ( - \infty , - 5] \cup [7,\infty ) is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of L 2 ( 1 / k ) {L_2}(1/k) . Moreover, the linear independence over Q {\mathbf {Q}} of the numbers 1 1 , log ⁡ ( 1 − 1 / k ) \log (1 - 1/k) , and L 2 ( 1 / k ) {L_2}(1/k) is also obtained for each integer k ∈ ( − ∞ , − 5 ] ∪ [ 7 , ∞ ) k \in ( - \infty , - 5] \cup [7,\infty ) .

Another Proof of the Irrationality of

Another Proof of the Irrationality of