The constant
that Gelfond's theorem established to be transcendental
seems to lack a generally accepted name. As a result, in this work, it will be dubbed
Gelfond's constant. Both the Gelfond-Schneider
constant
and Gelfond's constant were singled out in the 7th of Hilbert's
problems as examples of numbers whose transcendence was an open problem (Wells
1986, p. 45).
Gelfond's constant has the numerical value
(1)
(OEIS A039661 ) and simple
continued fraction
(2)
(OEIS A058287 ).
Its digits can be computed efficiently using the iteration
(3)
with ,
and then plugging in to
(4)
(Borwein and Bailey 2003, p. 137).
See also e ,
Gelfond-Schneider
Constant ,
Gelfond's Theorem ,
Pi
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References Berggren, L.; Borwein, J.; and Borwein, P. Pi: A Source Book. New York: Springer-Verlag, p. 422, 1997. Borwein,
J. and Bailey, D. Mathematics
by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, 2003. Gullberg, J. Mathematics
from the Birth of Numbers. New York: W. W. Norton, p. 86,
1997. Hilbert, D. "Mathematical Problems." Bull. Amer. Math.
Soc. 8 , 437-479, 1902. Reprinted in Bull. Amer. Math. Soc. 37 ,
407-436, 2000. Sloane, N. J. A. Sequences A039661
and A058287 in "The On-Line Encyclopedia
of Integer Sequences." Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 81, 1986. Referenced on Wolfram|Alpha Gelfond's Constant
Cite this as:
Weisstein, Eric W. "Gelfond's Constant."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/GelfondsConstant.html
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