%I A007507 M1560
%S A007507 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,
%T A007507 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,
%U A007507 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 A007507 Decimal expansion of 2^sqrt(2).
%C A007507 "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
%C A007507 This constant is sometimes called the Gelfond-Schneider constant. [From 
               Paul Muljadi (paulmuljadi(AT)yahoo.com), Oct 12 2008]
%D A007507 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007507 David Wells, "The Penguin Dictionary of Curious and Interesting Numbers,
               " Revised Edition, Penguin Books, London, England, 1997, page 28.
%D A007507 Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 
               2002, p. 1171. [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Oct 
               12 2008]
%H A007507 Harry J. Smith, <a href="b007507.txt">Table of n, a(n) for n=1,...,20000</
               a>
%H A007507 D. Hilbert, <a href="http://www.ams.org/journal-getitem?pii=S0273-0979-00-00881-8">
               Mathematical Problems</a>, Bull. Amer. Math. Soc. 37 (2000), 407-436. 
               Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437-479. See 
               Problem 7.
%H A007507 S. Plouffe, Plouffe's Inverter, <a href="http://pi.lacim.uqam.ca/piDATA/
               gelfond.txt">2**sqrt(2), a transcendental number to 5000 digits</
               a>
%H A007507 S. Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/
               math/MiscellaneousMathematicalConstants/chap43.html">2**sqrt(2), 
               a transcendental number to 2000 digits</a>
%H A007507 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Gelfond-SchneiderConstant.html">Link to a section of The World of 
               Mathematics</a>
%e A007507 2.6651441426902251886502972498731398482742113137146594928...
%t A007507 RealDigits[N[ 2^Sqrt[2], 100]][[1]]
%o A007507 (PARI) { default(realprecision, 20080); x=2^sqrt(2); for (n=1, 20000, 
               d=floor(x); x=(x-d)*10; write("b007507.txt", n, " ", d)); } [From 
               Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
%Y A007507 Sequence in context: A071678 A141329 A110388 this_sequence A065486 A069806 
               A123945
%Y A007507 Adjacent sequences: A007504 A007505 A007506 this_sequence A007508 A007509 
               A007510
%K A007507 cons,nonn
%O A007507 1,1
%A A007507 N. J. A. Sloane (njas(AT)research.att.com).
%E A007507 Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 07 
               2000
%E A007507 Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               May 19 2009
%E A007507 Final digits of sequence corrected using the b-file. - N. J. A. Sloane, 
               Aug 30 2009