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%I M1560 #42 Aug 25 2020 06:35:58
%S 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,
%T 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,
%U 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
%N Decimal expansion of 2^sqrt(2).
%C "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
%C This constant is sometimes called the Gelfond-Schneider constant. - _Paul Muljadi_, Oct 12 2008
%C From _Amiram Eldar_, Aug 25 2020: (Start)
%C The transcendence of this number was proved by the Russian mathematician Rodion Osievich Kuzmin (1891 - 1949) in 1930.
%C 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)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 28.
%D Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 2002, p. 1171.
%H Harry J. Smith, <a href="/A007507/b007507.txt">Table of n, a(n) for n = 1..20000</a>
%H Aleksandr Gelfond, <a href="http://mi.mathnet.ru/eng/izv4924">Sur le septième Problème de Hilbert</a>, Bulletin de l'Académie des Sciences de l'URSS, Classe des sciences mathématiques et na. VII, No. 4 (1934), pp. 623-634.
%H David Hilbert, <a href="https://doi.org/10.1090/S0273-0979-00-00881-8">Mathematical Problems</a>, 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.
%H R. O. Kuzmin, <a href="http://mi.mathnet.ru/eng/izv5316">On a new class of transcendental numbers</a>" (in Russian), Izvestiya Akademii Nauk SSSR, Ser. matem. 7, No. 6 (1930), pp. 585-597.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/gelfond.txt">2**sqrt(2), a transcendental number to 5000 digits</a>.
%H Simon Plouffe, <a href="https://web.archive.org/web/20080205202359/http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap43.html">2**sqrt(2), a transcendental number to 2000 digits</a>.
%H Theodor Schneider, <a href="https://eudml.org/doc/149900">Transzendenzuntersuchungen periodischer Funktionen I. Transzendenz von Potenzen</a>, J. reine angew. Math., Vol. 172 (1935), pp. 65-69.
%H Theodor Schneider, <a href="https://eudml.org/doc/149901">Transzendenzuntersuchungen periodischer Funktionen II. Transzendenzeigenschaften elliptischer Funktionen</a>, J. reine angew. Math., Vol. 172 (1934), pp. 70-74.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Gelfond-SchneiderConstant.html">Gelfond-Schneider Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant">Gelfond-Schneider constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem">Gelfond-Schneider theorem</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e 2.6651441426902251886502972498731398482742113137146594928...
%t RealDigits[N[ 2^Sqrt[2], 100]][[1]]
%o (PARI) default(realprecision, 20080); x=2^sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b007507.txt", n, " ", d)); \\
%K cons,nonn
%O 1,1
%A _N. J. A. Sloane_
%E Final digits of sequence corrected using the b-file. - _N. J. A. Sloane_, Aug 30 2009
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