1 Irrationality 3 1.1 Simple proofs of irrationality . . . . . . . . . . . . . . . . . . . . 3 1.2 Variation on a proof by Fourier (1815) . . . . . . . . . . . . . . . 10 1.2.1 Irrationality of e . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 The number e is not quadratic . . . . . . . . . . . . . . . 11 1.2.3 Irrationality of e √ 2 (Following a suggestion of D.M. Masser) . . . . . . . ...

1 Irrationality 3 1.1 Simple proofs of irrationality . . . . . . . . . . . . . . . . . . . . 3 1.1.1 History of irrationality . . . . . . . . . . . . . . . . . . . . 10 1.2 Variation on a proof by Fourier (1815) . . . . . . . . . . . . . . . 12 1.2.1 Irrationality of e . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.2 The number e is not quadratic . . . . . . . . . . . . . . . 13 1.2.3 Irr...

1 Irrationality 3 1.1 Simple proofs of irrationality . . . . . . . . . . . . . . . . . . . . 3 1.2 Variation on a proof by Fourier (1815) . . . . . . . . . . . . . . . 10 1.2.1 Irrationality of e . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 The number e is not quadratic . . . . . . . . . . . . . . . 11 1.2.3 Irrationality of e √ 2 (Following a suggestion of D.M. Masser) . . . . . . . ...

1. Introduction. In his proof of the irrationality of ζ(3), Apéry [1] gave sequences of rational approximations to ζ(2) = π 2 /6 and to ζ(3) yielding the irrationality measures µ(ζ(2)) < 11.85078. .. and µ(ζ(3)) < 13.41782. .. Several improvements on such irrationality measures were subsequently given, and we refer to the introductions of the papers [3] and [4] for an account of these results. ...

w x Ž . In 1948, Paul Erdos E1 proved the irrationality of h 1 . Recently, ̋ 2 w x Peter Borwein used Pade approximation techniques B1 and some coḿ w x Ž . plex analysis methods B2 to prove the irrationality of both h 1 and q Ž . Ln 2 . Here we present a proof in the spirit of Apery’s magnificent proof ́ q Ž . w x of the irrationality of z 3 A , which was later delightfully accounted by w x Alf v...

We demonstrate that also the second sum involved in Apéry’s proof of the irrationality of ζ(3) becomes trivial by symbolic summation. In his beautiful survey [4], van der Poorten explained that Apéry’s proof [1] of the irrationality of ζ(3) relies on the following fact: If a(n) = n

According to normative theories, reward-maximizing agents should have consistent preferences. Thus, when faced with alternatives A, B, and C, an individual preferring A to B and B to C should prefer A to C. However, it has been widely argued that humans can incur losses by violating this axiom of transitivity, despite strong evolutionary pressure for reward-maximizing choices. Here, adopting a ...

There are not many new results concerning the linear independence of numbers. Exceptions in the last decade are, e.g., the result of Sorokin [8] which proves the linear independence of logarithmus of special rational numbers, or that of Bezivin [2] which proves linear independence of roots of special functional equations. The algebraic independence of numbers can be considered as a generalizati...

The story exposed in this paper starts in 1978, when R. Apéry [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apéry’s discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apéry’s proof allows ...