Two IBM researchers, with the assistance of the present bloggers (Bailey and Borwein), have calculated binary digits of pi-squared beginning at the 60 trillionth binary digit. In addition, they calculated ternary (base-3) digits of pi-squared, beginning at the 60 trillionth digit, and binary digits of Catalan’s constant, beginning at the 120 trillionth digit. Other researchers have previously calculated binary digits of pi itself, beginning at the two quadrillionth binary digit.

These calculations would have taken a single computer processor unit (CPU) 1,500 years to calculate, but the researchers managed to complete this work in just a few months on IBM’s “BlueGene/P” supercomputer, which is designed to run continuously at one quadrillion calculations per second.

These calculations were based on some mathematical formulas published in a 1997 paper by David H. Bailey (one of the present bloggers), Peter Borwein (Jonathan’s brother), and Simon Plouffe of Canada. These formulas have the remarkable property that they permit one to calculate binary digits (or, in some cases, ternary digits or digits in other number bases) of various mathematical constants, beginning at an arbitrary position, without needing to calculate any of the digits than came before. These formulas were originally discovered by computer programs, using the “PSLQ” algorithm of sculptor-mathematician Helaman Ferguson. For the computation of digits of Catalan’s constant described here, an improved formula was found by the present authors.

Just for the record:

1. The first 12 base-8 digits of pi-squared beginning at position 20 trillion (corresponding to binary position 60 trillion) are: 601145053032.

2. The first 12 base-9 digits of pi-squared beginning at position 30 trillion (corresponding to ternary position 60 trillion) are: 122644850645.

3. The first 12 base-8 digits of Catalan’s constant beginning at position 40 trillion (corresponding to binary position 120 trillion) are: 347050537747.

Full details are available in this online manuscript: BBP-Bluegene paper.

A short history of pi is available on our earlier Expmath blog.

A news article on the calculation, published on the U.S. Department of Energy’s Office of Science website, is available here: DOE news article.