We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log(2) or π on a modest work station in a few hours run time.

/pdf/on-the-rapid-computation-of-various-polylogarithmic-2jkwpfausz.pdf

On the rapid computation of various polylogarithmic constants

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or π on a modest work station in a few hours run time.

/pdf/on-the-rapid-computation-of-various-polylogarithmic-3f509b3nqm.pdf

James Propp

The most familiar construction of graphs whose clique number is much smaller than their chromatic number is due to Mycielski, who constructed a sequence Gn of triangle-free graphs with X(Gn) = n. In this article, we calculate the fractional chromatic number of Gn and show that this sequence of numbers satisfies the unexpected recurrence an+1 = an + (1/an). © 1995 John Wiley & Sons, Inc.

The fractional chromatic number of Mycielski's graphs

This article gives a brief history of the analysis and computation of the mathematical constant Pi=3.14159 ..., including a number of the formulas that have been used to compute Pi through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of Pi to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of Pi to be computed.

The quest for PI

The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system. It has conjured up tremendous interest in mathematicians and non-mathematicians alike, who spent countless hours over millennia to explore its beauty and varied applications in science and engineering. The article attempts to record the pi exploration over centuries including its successive computation to ever increasing number of digits and its remarkable usages, the list of which is not yet closed.

/pdf/birth-growth-and-computation-of-pi-to-ten-trillion-digits-4prm0tto4a.pdf

Birth, growth and computation of pi to ten trillion digits

(1995). A Spigot Algorithm for the Digits of π. The American Mathematical Monthly: Vol. 102, No. 3, pp. 195-203.

A spigot algorithm for the digits of pi

Paper 13: Stanley Rabinowitz and Stan Wagon, “A spigot algorithm for the digits of π,” American Mathematical Monthly, vol. 102 (March 1995), p. 195–203. Copyright 1995 Mathematical Association of America. All Rights Reserved.

Stanley Rabinowitz

Papers

A spigot algorithm for the digits of pi

A spigot algorithm for the digits of π (1995)

Elementary Problems: E2043,E2095-E2104

Problems and Solutions: Problems: 10382-10388

Advanced Problems: 5636-5643