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Search: id:A007507
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| A007507 |
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Decimal expansion of 2^sqrt(2).
(Formerly M1560)
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+0
2
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| 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
(list; cons; graph; listen)
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OFFSET
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1,1
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COMMENT
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"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
This constant is sometimes called the Gelfond-Schneider constant. [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Oct 12 2008]
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REFERENCES
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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. [From Paul Muljadi (paulmuljadi(AT)yahoo.com), Oct 12 2008]
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LINKS
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Harry J. Smith, Table of n, a(n) for n=1,...,20000
D. Hilbert, Mathematical Problems, Bull. Amer. Math. Soc. 37 (2000), 407-436. Reprinted from Bull. Amer. Math. Soc. 8 (Jul 1902), 437-479. See Problem 7.
S. Plouffe, Plouffe's Inverter, 2**sqrt(2), a transcendental number to 5000 digits
S. Plouffe, 2**sqrt(2), a transcendental number to 2000 digits
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics
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EXAMPLE
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2.6651441426902251886502972498731398482742113137146594928...
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MATHEMATICA
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RealDigits[N[ 2^Sqrt[2], 100]][[1]]
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PROGRAM
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(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]
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CROSSREFS
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Sequence in context: A071678 A141329 A110388 this_sequence A065486 A069806 A123945
Adjacent sequences: A007504 A007505 A007506 this_sequence A007508 A007509 A007510
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KEYWORD
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cons,nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 07 2000
Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009
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