Gelfond's Constant

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).

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.