A new formula for picking off pieces of pi.

Members of the Ancient and Honorable Society of Pi Watchers have something new to gaze upon and ponder.

Mathematicians have discovered a surprisingly simple formula for computing digits of the number pi ([pi]). Unlike previously known expressions, this one allows them to calculate isolated digits--say, the billionth digit of pi--without computing and keeping track of all the preceding numbers.

"It was something that people just didn't think you could do," says Peter B. Borwein of Simon Fraser University in Burnaby, British Columbia.

The only catch is that the formula works for binary, but not for decimal, digits. Thus, it's possible to determine that the forty billionth binary digit of pi is 1, followed by 00100100001110. . . . But there's no way to convert these numbers into decimal form without knowing all the binary digits that come before the string.

Borwein, Simon Fraser colleague Simon Plouffe, and their coworkers announced the discovery earlier this month by posting it on the Internet.

"It's a curious finding," says Helaman Ferguson of the Supercomputing Research Center in Bowie, Md., who has checked the result. "It's quite surprising that this [formula] exists."

Borwein has long been interested in finding efficient ways of computing pi, the ratio of a circle's circumference to its diameter. In particular, he has focused on methods of performing the computation using only a small amount of computer memory.

Working with Plouffe, Borwein identified certain types of mathematical expressions, or series, that would provide the necessary shortcut. An extensive search turned up suitable formulas for pi and several other numerical constants, including log(2).

However, the answer that comes out of the expression for pi gives only hexadecimal (base 16) digits, which can be readily converted to binary. "The frustrating thing is that it doesn't work in base 10 [for decimal digits]," Borwein remarks.

Borwein and his coworkers are still hoping to uncover an expression that gives the decimal digits of pi, but other mathematicians are pessimistic that such a formula will ever be found. Meanwhile, the researchers have been looking for related series to compute other mathematical constants, such as e and the square root of 2, but with limited success so far.

The existence of such an intriguing formula for computing isolated digits of pi may reveal something mathematical about the nature of the number itself. For example, mathematicians would like to prove that all the decimal digits occur in pi equally often.

"That would be the mathematical prize in all this," Borwein notes. "But at the moment, I can't see [our discovery] leading to a proof."

On the decimal front, Yasumasa Kanada and his coworkers at the University of Tokyo have now computed pi to 4,294,960,000 digits, beating the old record (SN: 8/26/95, p.143). According to their calculations, the four billionth decimal digit of pi is 9, followed by 4375343. . . .

The researchers also show that in the first 4 billion digits, the number 6 appears most often (400,033,035 times) and 2 least often (399,965,405 times).

"We do the pi calculations from scratch because [this] is one of the best benchmark programs for testing the reliability of [our computers] and checking the correctness of calculations, programs, and algorithms," Kanada says. "To be a world record holder is a by-product."

If researchers ever find a decimal equivalent of the new formula for binary digits of pi, Kanada and others would be able to push their calculations much higher. Indeed, because such a formula would enable them to compute isolated clumps of digits, the task could be readily divided up among as many computers as necessary to get the desired result.

Members of Pi Watchers and others interested in the new formula for the rapid computation of pi can obtain additional information from Borwein's World Wide Web site at the following address: http://www.cecm.sfu.ca/~pborwein/. Anyone curious about 4 billion decimal digits of pi can check Kanada's site at: http://www.cc.u-tokyo.ac.jp/.