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A007507 Decimal expansion of 2^sqrt(2).
(Formerly M1560)
3
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; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

This constant is sometimes called the Gelfond-Schneider constant. - Paul Muljadi, Oct 12 2008

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. [From Paul Muljadi, Oct 12 2008]

LINKS

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.

Simon Plouffe, 2**sqrt(2), a transcendental number to 5000 digits

Simon Plouffe, 2**sqrt(2), a transcendental number to 2000 digits

Eric Weisstein's World of Mathematics, Gelfond-Schneider Constant

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)); } \\ Harry J. Smith, Apr 21 2009

CROSSREFS

Sequence in context: A071678 A141329 A110388 * A065486 A069806 A123945

Adjacent sequences:  A007504 A007505 A007506 * A007508 A007509 A007510

KEYWORD

cons,nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Robert G. Wilson v, Dec 07 2000

Fixed my PARI program, had -n Harry J. Smith, May 19 2009

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

STATUS

approved

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Last modified March 30 18:30 EDT 2017. Contains 284302 sequences.