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Gelfond's Constant

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The constant e^pi 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 2^(sqrt(2)) and Gelfond's constant e^pi 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

e^pi=23.140692632...
(1)

(OEIS A039661) and simple continued fraction

e^pi=[23,7,9,3,1,1,591,2,9,1,2,34,...]
(2)

(OEIS A058287).

Its digits can be computed efficiently using the iteration

k_n=(1-sqrt(1-k_(n-1)^2))/(1+sqrt(1-k_(n-1)^2))
(3)

with k_0=1/sqrt(2), and then plugging in to

e^pi approx (1/4k_n)^(-2^(1-n))
(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.

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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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