Computation of the regular continued fraction for Euler's constant
40. R. P. Brent,
Computation of the regular continued fraction for Euler's constant,
Mathematics of Computation 31 (1977), 771-777.
MR 55\#9490.
Abstract:
dvi (3K),
pdf (82K),
ps (28K).
Paper:
pdf (455K).
Review by Daniel Shanks in
Mathematics of Computation 32 (1978), 311:
pdf (107K).
Abstract
We describe a computation,
using the MP package,
of the first 20,000 partial quotients in
the regular continued fraction for Euler's constant
gamma = 0.577...
and exp(gamma) = 1.781...
A preliminary step was the calculation
of gamma and exp(gamma) to 20,700D.
It follows from the continued
fractions that, if gamma or exp(gamma) is of the form
P/Q for
integers P and Q, then
|Q| > 1010000.
Comments
It is not known whether gamma or exp(gamma) is rational;
the lower bound on Q shows that neither is a
small rational.
The computation of gamma improved on an earlier
(and only partially correct)
result of Beyer and Waterman
[Mathematics of Computation 28 (1974), 599-604].
An interesting connection with the work of
Ramanujan
is described in Brent [139].
For a sequel which introduced some more efficient algorithms
and extended the computation, see
Brent and McMillan [49].
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