The evolution of extended decimal approximations to π

Wrench, John W.

doi:10.1007/978-1-4757-4217-6_37

Cited by 2 publications

(2 citation statements)

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“…Consequently -and in contrast to p value inference-Bayes factors can be seamlessly updated as new data arrive, indefinitely and without a well-defined sampling plan Berry, 1988a, Berger andBerry, 1988b]. This property is particularly relevant for the study of normality of fundamental constants, since new computational and mathematical developments continually increase the length of the decimal expansion [Wrench Jr, 1960].…”

Section: Bayes Factors To Quantify Evidence For General Lawsmentioning

confidence: 99%

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Gronau

Wagenmakers

2017

Experimental Mathematics

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Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference -the logic of partial beliefs-can be used to quantify the evidence that finite data provide in favor of a general law. As a concrete example we focus on the general law which posits that certain fundamental constants (i.e., the irrational numbers π, e, √ 2, and ln 2) are normal; specifically, we consider the more restricted hypothesis that each digit in the constant's decimal expansion occurs equally often. Our analysis indicates that for each of the four constants, the evidence in favor of the general law is overwhelming. We argue that the Bayesian paradigm is particularly apt for applications in experimental mathematics, a field in which the plausibility of a general law is in need of constant revision in light of data sets whose size is increasing continually and indefinitely.

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Section: Bayes Factors To Quantify Evidence For General Lawsmentioning

confidence: 99%

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Gronau

Wagenmakers

2017

Experimental Mathematics

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“…The problem of expressing π as the sum of arctangents has been deeply studied during the years. The first expressions are due to Newton (1676), Machin (1706), Euler (1755), who expressed π using the following identities respectively (see, e.g., [12] and [13]). Many other identities and methods to express and calculate π involving the arctangent function have been developed.…”

Section: Expressions Of π Via Arctangent Function With Linear Recurrementioning

confidence: 99%

Writing π as sum of arctangents with linear recurrent sequences, Golden mean and Lucas numbers

Abrate

Barbero

Cerruti

et al. 2014

Int. J. Number Theory

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The evolution of extended decimal approximations to π

Cited by 2 publications

References 25 publications

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

Writing π as sum of arctangents with linear recurrent sequences, Golden mean and Lucas numbers

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