A007507 - OEIS
2, 6, 6, 5, 1, 4, 4, 1, 4, 2, 6, 9, 0, 2, 2, 5, 1, 8, 8, 6, 5, 0, 2, 9, 7, 2, 4, 9, 8, 7, 3, 1, 3, 9, 8, 4, 8, 2, 7, 4, 2, 1, 1, 3, 1, 3, 7, 1, 4, 6, 5, 9, 4, 9, 2, 8, 3, 5, 9, 7, 9, 5, 9, 3, 3, 6, 4, 9, 2, 0, 4, 4, 6, 1, 7, 8, 7, 0, 5, 9, 5, 4, 8, 6, 7, 6, 0, 9, 1, 8, 0, 0, 0, 5, 1, 9, 6, 4, 1, 6, 9, 4, 1, 9, 8
COMMENTS
"The 7th of Hilbert's famous 23 problems proposed at the 1900 Mathematical Congress was to prove the irrationality and transcendence of certain numbers. Hilbert gave as examples 2^sqrt(2) and e^Pi. Later in his life he expressed the view that this problem was more difficult than the problems of Riemann's hypothesis or Fermat's Last Theorem. Nevertheless, e^Pi was proved transcendental in 1929 and 2^sqrt(2) in 1930, illustrating the extreme difficulty of anticipating the future progress of mathematics and the real difficulty of any problem - until after it has been solved." - David Wells - Robert G. Wilson v, Dec 07 2000
This constant is sometimes called the Gelfond-Schneider constant. - Paul Muljadi, Oct 12 2008
The transcendence of this number was proved by the Russian mathematician Rodion Osievich Kuzmin (1891 - 1949) in 1930.
It was named after the Soviet mathematician Alexander Osipovich Gelfond (1906 - 1968) and the German mathematician Theodor Schneider (1911 - 1988), who independently proved the Gelfond-Schneider theorem from which the transcendence of this number follows. (End)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 28.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002, p. 1171.
LINKS
Aleksandr Gelfond, Sur le septième Problème de Hilbert, Bulletin de l'Académie des Sciences de l'URSS, Classe des sciences mathématiques et na. VII, No. 4 (1934), pp. 623-634.
David Hilbert, Mathematical Problems, Bull. Amer. Math. Soc., Vol. 37, No. 4 (2000), pp. 407-436. Reprinted from Bull. Amer. Math. Soc., Vol. 8, No. 10 (1902), pp. 437-479. See Problem 7.
EXAMPLE
2.6651441426902251886502972498731398482742113137146594928...
MATHEMATICA
RealDigits[N[ 2^Sqrt[2], 100]][[1]]
PROG
(PARI) default(realprecision, 20080); x=2^sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b007507.txt", n, " ", d)); \\
EXTENSIONS
Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009