Favorite indoor sports: Search for the "best" arctan relation

Joerg Arndt (MSI, ANU)

MSI Computational Mathematics Seminar Series

DATE: 2007-05-14
TIME: 11:00:00 - 12:00:00
LOCATION: John Dedman G35
CONTACT: JavaScript must be enabled to display this email address.

ABSTRACT:
A computer search for relations of the form M1*arctan(1/X1) + M2*arctan(1/X2) + ... + Mn*arctan(1/Xn) = k*Pi/4 is described. We search for the "best" arctan n-term relation, the one were the smallest Xi is a big as possible. For example, the following 5-term relation is found:

Pi/4 = + 88 arctan(1/192) + 39 arctan(1/239) + 100 arctan(1/515) - 32 arctan(1/1068) - 56 arctan(1/173932)

For all (inverse) arguments X the prime factors of X^2 +1 are in the set {2,5,13,73,101}. A fast algorithm is given to determine all X in a prescribed range so that X^2 + 1 factors into a given set of primes. The resulting table of values X <= 10^14 that factor into primes p <= 761 is used to rediscover all previously known "good" relations and find new relations with up to 27 terms.

The search was possible only with highly optimized methods that will be explained.

The best 2-term, 3-term and 4-term relations were known before 1900 (the relations were given by Machin, Gauss, and Stormer). All records for 5-term ... 27-term relations (with the exception of the 15-term relation which is by Hwang Chien-lih) were obtained in my 1993 and 2006 computations.

A warning about the addictiveness of the topic will be given.