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This issue...

  News in Brief

  SciDAC

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  New Lens Helps Find Cancer Tumors

  Are the Digits of Pi Random?

  Doubly Strange Nuclei

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  Subscribe Free

Are the Digits of Pi Random?

A Berkeley Lab Researcher May Hold the Key

by Paul Preuss

A researcher at the Department of Energy's National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory, and his colleague at the Center for Advanced Computation at Reed College, have taken a major step toward answering the age-old question of whether the digits of pi and other math constants are "random." Their results are reported in the Summer 2001 issue of Experimental Mathematics.

Pi symbolPi, the ubiquitous number whose first few digits are 3.14159, is irrational, which means that its digits run on forever (by now they have been calculated to billions of places) and never repeat in a cyclical fashion. Numbers like pi are also thought to be "normal," which means that their digits are random in a certain statistical sense.

David Bailey
David Bailey
Describing the normality property, David H. Bailey, chief technologist at NERSC, explains that "in the familiar base 10 decimal number system, any single digit of a normal number occurs one tenth of the time, any two-digit combination occurs one one-hundredth of the time, and so on. It's like throwing a fair, ten-sided die forever and counting how often each side or combination of sides appears."

Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.

In fact, not a single naturally occurring math constant has been proved normal in even one number base, to the chagrin of mathematicians. While many constants are believed to be normal—including pi, the square root of 2, and the natural logarithm of 2, often written "log(2)"—there are no proofs.

The determined attacks of Bailey and his colleague Richard Crandall, director of the Center for Advanced Computation at Reed College, Portland, Oregon, are beginning to illuminate this classic problem. Their results indicate that the normality of certain math constants is a consequence of a plausible conjecture in the field of chaotic dynamics, which states that sequences of a particular kind, as Bailey puts it, "uniformly dance in the limit between 0 and 1"—a conjecture that he and Crandall refer to as "Hypothesis A."

"If even one particular instance of Hypothesis A could be established," Bailey remarks, "the consequences would be remarkable"—for the normality (in base 2) of pi and log(2) and many other mathematical constants would follow.

Pi algorithm


A simple formula discovered with the integer-relation algorithm dubbed PSLQ makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power.

This result derives directly from the discovery of an ingenious formula for pi that Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found with a computer program in 1996. Named the BBP formula for its authors, it has the remarkable property that it permits one to calculate an arbitrary digit in the binary expansion of pi without needing to calculate any of the preceding digits. Prior to 1996, mathematicians did not believe this could be done.

The digit-calculation algorithm of the BBP formula yields just the kind of chaotic sequences described in Hypothesis A. Says Bailey, "These constant formulas give rise to sequences that we conjecture are uniformly distributed between 0 and 1—and if so, the constants are normal."

Bailey emphasizes that the new result he and Crandall have obtained does not constitute a proof that pi or log(2) is normal (since this is predicated on the unproven Hypothesis A). "What we have done is translate a heretofore unapproachable problem, namely the normality of pi and other constants, to a more tractable question in the field of chaotic processes."

He adds that "at the very least, we have shown why the digits of pi and log(2) appear to be random: because they are closely approximated by a type of generator associated with the field of chaotic dynamics."

For the two mathematicians, the path to their result has been a long one. Bailey memorized pi to more than 300 digits "as a diversion between classroom lectures" while still a graduate student at Stanford. In 1985 he tested NASA's new Cray-2 supercomputer by computing the first 29 million digits of pi. The program found bugs in the Cray-2 hardware, "much to the consternation of Seymour Cray."

Crandall, who researches scientific applications of computation, suggested the possible link between the digits of pi and the theory of chaotic dynamic sequences.

While other prominent mathematicians in the field fear that the crucial Hypothesis A may be too hard to prove, Bailey and Crandall remain sanguine. Crandall quotes the eminent mathematician Carl Ludwig Siegel: "One cannot guess the real difficulties of a problem before having solved it."

Among the numerous connections of Bailey's and Crandall's work with other areas of research is in the field of pseudorandom number generators, which has applications in cryptography.

"The connection to pseudorandom number generators is likely the best route to making further progress," Bailey adds. "Richard and I are pursuing this angle even as we speak."

Media contact: Paul Preuss, (510) 486-6249, paul_preuss@lbl.gov
Research contact: David Bailey, (510) 495-2773, dhbailey@lbl.gov


Related Web Links

For more about the normality of pi and other constants, visit David Bailey's website at http://www.nersc.gov/~dhbailey.

The BBP algorithm for calculating binary digits of pi was found using the PSLQ algorithm developed by Bailey and mathematician-sculptor Helaman Ferguson; it is also discussed at Bailey's website.

"Algorithm for the Ages: Better Way to Find Integer Relations," Paul Preuss, Berkeley Lab Research News, January 20, 2000.

Experimental Mathematics journal website

The Joy of Pi, a book by David Blatner. This website has links to more information and books about Pi research.


Funding: The National Energy Research Super Computing Center (NERSC) is funded by the Department of Energy's Office of Science. NERSC is a world leader in accelerating scientific discovery through computation. It provides high-performance computing tools and expertise that enable computational science of scale, in which large, interdisciplinary teams of scientists attack fundamental problems in science and engineering that require massive calculations and have broad scientific and economic impacts. NERSC's Dr. David Bailey is leading a multi-institution, DOE-funded research program in performance analysis and modeling for scientific computing.

The Lawrence Berkeley National Laboratory is a U.S. Department of Energy national laboratory located in Berkeley, California. It conducts unclassified scientific research and is managed by the University of California.

Author: Paul Preuss has been a science writer at Lawrence Berkeley National Laboratory for the last four years, covering the broad range of research conducted by the Lab, from astrophysics to earth sciences to chemistry and materials sciences to genetics to cell biology, etc. Before joining LBNL, Preuss spent 20 years as a novelist, writing mostly science fiction, even collaborating with Arthur C. Clarke on his "Venus Prime" series, which is still prominent on Amazon.com. Preuss said, "I've stopped writing science fiction. The truth is that real science is far more fascinating. The fictional stuff is a pale reflection."

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"What we have done is translate a heretofore unapproachable problem, the normality of Pi and other constants, to a more tractable question in the field of chaotic processes." —David Bailey
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