Biting off a record-breaking piece of pi.

Biting off a record-breaking piece of pi

"Compute but verify" is the strategy at the core of a remarkable new method for calculating pi ([pi])--the ratio of a circle's circumference to its diameter. In a recent demonstration of the method's power, two mathematicians at Columbia University in New York City used it to compute pi to 480 million decimal places, shattering the previous record of 201 million digits (SN: 4/2/88, p. 215).

"Our goal is to develop a better understanding of the arithmetic properties of [constants such as pi]," says David V. Chudnovsky, who with his brother Gregory discovered the key formulas for the computation. With so many digits of pi now available for analysis, the Chudnovskys are seeing the first hints of subtle patterns in the distribution of pi's digits, suggesting the digits may not be truly random.

Computing pi also gives the largest and fastest computers a thorough workout, pinpointing subtle flaws in their hardware and software. "It is really the ultimate stress test--a cardiogram for a computer," Chudnovsky says. In the course of their calculations, the Chudnovskys identified unexpected quirks peculiar to the computers they used.

The new formula, or identity, discovered and used by the Chudnovskys expresses pi as a complicated sum. By evaluating more and more terms in such a sum, mathematicians get closer and closer to the true value of pi. This particular identity gets closer to pi faster than any other known formula.

But computing pi quickly isn't enough. Computers don't work correctly all the time, and an accidental change in even one bit of data could completely corrupt a computation. "The problem [of hardware faults] is totally beyond your control," Chudnovsky says. "If you don't have a way immediately to verify and recover the data, you would lose everything, and you wouldn't know it." To avoid such difficulties, the Chudnovsky method includes automatic verification and, where necessary, correction.

Using computer programs written in FORTRAN, the Chudnovskys tested their method on a Cray-2 and an IBM 3090. They computed at odd times over a period of 6 months, sometimes in segments only 15 minutes long.

The Chudnovsky method lends itself to group efforts. Much of the work of computing pi to a given number of decimal places can readily be divided among a large number of people, each working independently on a small computer. The Chudnovskys envision a "pi chain letter," with interested researchers, students and hackers combining their efforts to do multibillion-digit calculations.

Computing pi to billions of digits is important in the search for patterns among pi's digits. "Even a billion digits aren't really enough for doing a proper statistical analysis," Chudnovsky says. Although early results show some evidence for subtle relationships among the numbers, "we don't have enough numerical evidence yet."