1. INTRODUCTION. While there exist geometric proofs of irrationality for √2 [2], [27], no such proof for e, π , or ln 2 seems to be known. In section 2 we use a geometric construction to prove that e is irrational. The proof leads in section 3 to a new measure of irrationality for e, that is, a lower bound on the distance from e to a given rational number, as a function of its denominator. A co...

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 ...

Agents that interact with humans are known to benefit from integrating behavioral science and exploiting the fact that humans are irrational. Therefore, when designing agents for interacting with automated agents, it is crucial to know whether the other agents are acting irrationally and if so to what extent. However, little is known about whether irrationality is found in automated agent desig...

Apéry’s proof [13] of the irrationality of ζ(3) is now over 25 years old, and it is perhaps surprising that his methods have not yielded any significant new results (although further progress has been made on the irrationality of zeta values [1, 14]). Shortly after the initial proof, Beukers produced two elegant reinterpretations of Apéry’s arguments; the first using iterated integrals and Lege...

Apéry’s proof [11] of the irrationality of ζ(3) is now over 25 years old, and it is perhaps surprising that his methods have not yielded any significantly new results (although further progress has been made on the irrationality of zeta values [1], [12]). Shortly after the initial proof, Beukers produced two elegant reinterpretations of Apéry’s arguments; the first using iterated integrals and ...

Most people strongly believe they are just, virtuous, and moral; yet regard the average person as distinctly less so. This invites accusations of irrationality in moral judgment and perception-but direct evidence of irrationality is absent. Here, we quantify this irrationality and compare it against the irrationality in other domains of positive self-evaluation. Participants (N = 270) judged th...

1. Introduction. The irrationality of values of the zeta-function ζ(s) at odd integers s ≥ 3 is one of the most attractive problems in number theory. Inspite of a deceptive simplicity and more than two-hundred-year history of the problem, all done in this direction can easily be counted. It was only 1978, when Apéry [A] obtained the irrationality of ζ(3) by a presentation of " nice " rational a...

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, we show that there exists a four-term real linear recurrence relation whose solutions allow us to produce a tractable equivalent criterion for the quadratic irrationality of ζ(3), where ζ is the classic Zeta function of Riemann or, for that matter, the quadratic...