Proof Of Irrationality Research Articles

A simpler elementary proof that e is irrational

In this short note I present an elementary proof of irrationality for the number e, the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that e is a rational number from the beginning.

Algorithm-assisted discovery of an intrinsic order among mathematical constants

In recent decades, a growing number of discoveries in mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces. As computers become more powerful, an intriguing possibility arises-the interplay between human intuition and computer algorithms can lead to discoveries of mathematical structures that would otherwise remain elusive. Here, we demonstrate computer-assisted discovery of a previously unknown mathematical structure, the conservative matrix field. In the spirit of the Ramanujan Machine project, we developed a massively parallel computer algorithm that found a large number of formulas, in the form of continued fractions, for numerous mathematical constants. The patterns arising from those formulas enabled the construction of the first conservative matrix fields and revealed their overarching properties. Conservative matrix fields unveil unexpected relations between different mathematical constants, such as π and ln(2), or e and the Gompertz constant. The importance of these matrix fields is further realized by their ability to connect formulas that do not have any apparent relation, thus unifying hundreds of existing formulas and generating infinitely many new formulas. We exemplify these implications on values of the Riemann zeta function ζ (n), studied for centuries across mathematics and physics. Matrix fields also enable new mathematical proofs of irrationality. For example, we use them to generalize the celebrated proof by Apéry of the irrationality of ζ (3). Utilizing thousands of personal computers worldwide, our research strategy demonstrates the power of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.

Sequence Transformations in Proofs of Irrationality of Some Fundamental Constants

Sequence Transformations in Proofs of Irrationality of Some Fundamental Constants

Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéry

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler–Mascheroni constant, the Catalan constant, or \(\zeta (5)\). Inspired by Frits Beukers’ elegant rendition of Apéry’s seminal proofs of the irrationality of \(\zeta (2)\) and \(\zeta (3)\), and heavily using algorithmic proof theory, we systematically searched for other similar integrals that lead to irrationality proofs. We found quite a few candidates for such proofs, including the square-root of \(\pi \) times \(\Gamma (7/3)/\Gamma (-1/6)\) and \(\Gamma (19/6)/\Gamma (8/3)\) divided by the square-root of \(\pi \).

On an Integral Identity

We give three elementary proofs of a nice equality of definite integrals, recently proven by Ekhad, Zeilberger, and Zudilin. The equality arises in the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences.

On the Irrationality and Transcendence of Rational Powers of e

A number that can’t be expressed as the ratio of two integers is called an irrational number. Euler and Lambert were the first mathematicians to prove the irrationality and transcendence of e. Since then there have been many other proofs of irrationality and transcendence of e and generalizations of that proof to rational powers of e. In this article we review various proofs of irrationality and transcendence of rational powers of e founded by mathematicians over the time.

Преобразование последовательностей в доказательствах иррациональности некоторых фундаментальных констант

Transformation of number sequences (convergence acceleration) is one of the classical chapters of numerical analysis. These algorithms are used both for solution of practical problems and for the development of more advanced numerical methods. At the same time, numerical methods have found numerous applications in the number theory. One of the classical problems of number theory is the proof of irrationality of some fundamental constants, where the high rate of convergence of sequences of rational numbers plays a crucial role. However, as far as we know, the applications of (classical) convergence acceleration algorithms to the proofs of irrationality do not exist. This study is an attempt to fill this gap and to draw attention to this direction of research.

Automatic discovery of irrationality proofs and irrationality measures

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 sometimes there is still need for a human touch.

Irrational Thoughts

SummaryThis historical article discusses the best proof of irrationality of roots of integers.

102.44 Another proof of the irrationality of square roots

102.44 Another proof of the irrationality of square roots

Sequences, modular forms and cellular integrals

AbstractIt is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

102.21 An addendum to Estermann's proof of the irrationality of

102.21 An addendum to Estermann's proof of the irrationality of

Degenerations of Gushel\u2013Mukai fourfolds, with a view towards irrationality proofs

We study a certain class of degenerations of Gushel–Mukai fourfolds as conic bundles, which we call tame degenerations and which are natural if one wants to prove that very general Gushel–Mukai fourfolds are irrational using the degeneration method due to Voisin, Colliot-Thélène–Pirutka, Totaro et al. However, we prove that no such tame degenerations exist.

Some Transcendence Results from a Harmless Irrationality Theorem

The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values for all non-zero algebraic values of $z$ ($z \ne 1$ in the case of $\ln{z}$). On the other hand, not even an irrationality proof is known for some numbers like $\,e^e$, $\,\pi^e$, $\,\pi^\pi$, $\,\ln{\pi}$, $\,\pi + e\,$ and $\,\pi \, e$, though it is well-known that at least one of the last two numbers is irrational. In this note, I first derive a more general form of this last result, showing that at least one of the sum and product of any two transcendental numbers is transcendental. I then use this to show that, given any complex number $\,t \ne 0, 1/e$, at least two of the numbers $\,\ln{t}$, $\,t + e\,$ and $\,t \, e\,$ are transcendental. I also show that $\,\cosh{z}$, $\sinh{z}\,$ and $\,\tanh{z}\,$ return transcendental values for all $\,z = r \, \ln{t}$, $\,r \in \mathbb{Q}$, $r \ne 0$. Finally, I use a recent algebraic independence result by Nesterenko to show that, for all integer $\,n > 0$, $\,\ln{\pi}\,$ and $\,\sqrt{n} \, \pi\,$ are linearly independent over $\mathbb{Q}$.

Judging Inappropriateness in Actions Expressing Emotion: A Feminist Perspective

Actions expressing strong emotions such as anger can be appropriate responses when an agent judges a serious injustice to have been committed. Certainly, a woman can experience these conditions and express herself through actions such as gesturing aggressively, gritting her teeth, or lashing out verbally. If she is consequently labeled “crazy,” “hysterical,” or “a bitch,” what has gone awry? This paper offers an analysis of the common charge of inappropriateness in the case of women’s actions expressing emotion. To begin, I will present core normative distinctions that define appropriate emotional expression. Following this, the “double-bind” of women’s actions expressing emotion will be explored with reference to the conflicting normative practices outlined in the first section of the paper. Put briefly, when a female agent surpasses gendered behavioral expectations, she is seen as having failed what can be called the first test of social coping. The perception of this failure shuts down further avenues for interpreting her behavior. Instead, the social inappropriateness of her emotion is used as further proof of irrationality. The arguments of the second section leave no doubt that gendered norms in the case of actions expressing emotions must be rejected both on epistemological and moral grounds. The final section of the paper explores epistemically and ethically viable alternatives for deciding the rational appropriateness of actions expressing emotion.

On the values of $G$-functions

In this paper we study the set $\mathbf{G}$ of values at algebraic points of analytic continuations of $G$-functions (in the sense of Siegel). This subring of $\mathbb{C}$ contains values of elliptic integrals, multiple zeta values, and values at algebraic points of generalized hypergeometric functions ${p+1}F\_p$ with rational coefficients. Its group of units contains non-zero algebraic numbers, $\pi$, $\Gamma(a/b)^b$ and $B(x,y)$ (with $a,b\in \mathbb{Z}$ such that $a/b\not\in \mathbb Z$, and $x,y\in\mathbb{Q}$ such that $B(x,y)$ exists and is non-zero). We prove that for any $\xi \in \mathbf{G}$, both $\operatorname{Re} \xi$ and $\mathrm{Im} , \xi$ can be written as $f(1)$, where $f$ is a $G$-function with rational coefficients of which the radius of convergence can be made arbitrarily large. As an application, we prove that quotients of elements of $\mathbf{G} \cap \mathbb{R}$ are exactly the numbers which can be written as limits of sequences $a\_n/b\_n$, where $\sum{n=0}^{\infty} a\_n z^n$ and $\sum\_{n=0}^{\infty} b\_n z^n$ are $G$-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for $\zeta(3)$, and gives answers to questions asked by T. Rivoal in “Approximations rationnelles des valeurs de la fonction Gamma aux rationnels: le cas des puissances”, Acta Arith. 142 (2010), no. 4, 347–365.

A simpler proof of a Katsurada’s theorem and rapidly converging series for $$\zeta {(2n+1)}$$ ζ ( 2 n + 1 ) and $$\beta {(2n)}$$ β ( 2 n )

In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $\beta{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Ap\'{e}ry constant, $\zeta(3)$, and the Catalan constant, $G = \beta(2)$. Interestingly, our series for $\,\zeta(3)\,$ converges faster than that used by Ap\'{e}ry in his irrationality proof (1978). Also, our series for $\,G\,$ converges faster than a celebrated one discovered by Ramanujan (1915). At last, I present a simpler, more direct proof for a recent theorem by Katsurada which generalizes the above results.

A simple series representation for Apéry's constant

Apéry's constant is the value of ζ (3) where ζ is the Riemann zeta function. ThusThis constant arises in certain mathematical and physical contexts (in physics for example ζ (3) arises naturally in the computation of the electron's gyromagnetic ratio using quantum electrodynamics) and has attracted a great deal of interest, not least the fact that it was proved to be irrational by the French mathematician Roger é and named after him. See [1,2].Numerous series representations have been obtained for ζ (3) many of which are rather complicated [3]. é used one such series in his irrationality proof. It is not known whether ζ (3) is transcendental, a question whose resolution might be helped by a study of an appropriate series representation of ζ (3).

Irrational Square Roots

SummaryIf students are presented the standard proof of irrationality of√2, can they generalize it to a proof of the irrationality of √p, p a prime if, instead of considering divisibility by p, they cling to the notions of even and odd used in the standard proof?

Recurrent Proofs of the Irrationality of Certain Trigonometric Values

(2010). Recurrent Proofs of the Irrationality of Certain Trigonometric Values. The American Mathematical Monthly: Vol. 117, No. 4, pp. 360-362.