The computation of previously inaccessible digits of π² and Catalan's constant

We recently concluded a very large mathematical calculation, uncovering objects that until recently were widely considered to be forever inaccessible to computation. Our computations stem from the “BBP” formula for π, which was discovered in 1997 using a computer program implementing the “PSLQ” integer relation algorithm. This formula has the remarkable property that it permits one to directly calculate binary digits of π, beginning at an arbitrary position d, without needing to calculate any of the first d - 1 digits. Since 1997 numerous other BBP-type formulas have been discovered for various mathematical constants, including formulas for π² (both in binary and ternary bases) and for Catalan’s constant. In this article we describe the computation of base-64 digits of π², base-729 digits of π², and base-4096 digits of Catalan’s constant, in each case beginning at the ten trillionth place, computations that involved a total of approximately 1:549 x 1019 floating-point operations. We also discuss connections between BBP-type formulas and the age-old unsolved questions of whether and why constants such as π; π²; log 2, and Catalan’s constant have “random” digits.