A number of expansions to the geometric proof of the irrationality of the square root of two have been put forward in the paper “Irrationality From The Book” by Steven J. Miller and David Montague. There are a number of other conceptually simple expansions which can be attempted. This paper shows some basic work towards proving the irrationality of √ p, where p is a prime, and of 3 √ 2. Althoug...

We give a proof of the irrationality of the p-adic zeta-values ζp(k) for p = 2, 3 and k = 2, 3. Such results were recently obtained by F.Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show irrationality of some other p-adic L-series values, and values of the p-adi...

produces ‘good’ rational approximations to log a. There are several ways of perfoming integration in (1) in order to show that the integrals lies in Q log a + Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and ...

produces ‘good’ rational approximations to log a. There are several ways of performing integration in (1) in order to show that the integral lies in Q log a+ Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and l...

Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős [9]. In 1991–1992 Peter Borwein [4] [5] used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to f...

Consensus forecasts are ine¢ cient. The weight placed on new information re‡ects the choices of the underlying individual forecasters— whose new information is noisier individually than in aggregate— so that the consensus forecast overweights older information. Using a cross-country panel of growth forecasts and new methodological insights, this paper …nds that: consensus forecasts are ine¢ cie...

In 1978, Apéry [2] proved the irrationality of ζ(3) by constructing two explicit sequences of integers (un)n and (vn)n such that 0 6= unζ(3)− vn → 0 and un → +∞, both at geometric rates. He also deduced from this an upper bound for the irrationality exponent μ(ζ(3)) of ζ(3). In general, the irrationality exponent μ(ξ) of an irrational number ξ is defined as the infimum of all real numbers μ suc...

In 1978, R. Apéry [1], [2] has given sequences of rational approximations to ζ(2) and ζ(3) yielding the irrationality of each of these numbers. One of the key ingredient of Apéry’s proof are second-order difference equations with polynomial coefficients satisfied by numerators and denominators of the above approximations. Recently, V.N. Sorokin [3] and this author [4], [5] have got independentl...

1. Irrationality Measures An irrationality measure of x ∈ R \Q is a number μ such that ∀ > 0,∃C > 0,∀(p, q) ∈ Z, ∣∣∣∣x− pq ∣∣∣∣ ≥ C qμ+ . This is a way to measure how well the number x can be approximated by rational numbers. The measure is effective when C( ) is known. We denote inf {μ | μ is an irrationality measure of x } by μ(x), and we call it the irrationality measure of x. By definition,...

0. In 1978, Apéry showed the irrationality of ζ(3) = ∑∞ n=1 1 n3 by giving the approximants `n = unζ(3) − vn ∈ Qζ(3) + Q, un, dnvn ∈ Z, dn = l.c.m.(1, 2, . . . , n), with the property |`n| → ( √ 2 − 1) < 1/e as n → ∞. A similar approach was put forward to show the irrationality of ζ(2) (which is π/6, hence transcendental thanks to Lindemann) but I will concentrate on the case of ζ(3). A few mon...