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David
Bailey |
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When
the January/February 2000 issue of Computing
in Science and Engineering magazine named their
top 10 “Algorithms of the Century,” the list
included the integer-relation algorithm PSLQ,
discovered by mathematician and sculptor Helaman
Ferguson of Maryland’s Center for Computing Sciences,
and implemented in practical computer software
by David Bailey, NERSC’s chief technologist.
As
a tool of experimental mathematics, PSLQ’s purpose is to discover
new mathematical relations among numbers. In a short time it has
found polylogarithmic formulas in algebraic number theory, identified
a class of multiple-sum constants, uncovered relations in the renormalization
procedures of quantum field theory symbolized by Feynman diagrams,
and — most surprisingly — found a formula for calculating any digit
of  without calculating
the digits preceding it.
Before
PSLQ, mathematicians had not thought that such a digit-extraction
algorithm for
was possible. Using the remarkably simple formula, even a personal
computer can calculate ’s
millionth binary digit in about 60 seconds. Most applications of
PSLQ, however, require much more computing power and must employ
much greater numerical precision than the standard 16-digit, 64-bit,
fioating-point arithmetic available on most computers. That’s why
Bailey has developed software that translates ordinary C or Fortran
programs into programs capable of arbitrary precision — calculations
accurate to tens, hundreds, or even many thousands of digits.
Some
of PSLQ’s results have profound implications.
The
formula raises questions about the long-held
but never proved assumption that ’s
digits are random. The Feynman-diagram results
hint at unsuspected relationships among formulas
associated with fundamental particles. These
discoveries suggest that experimental mathematics
using computers will become increasingly important
in this new century.
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