A007507 Decimal expansion of 2^sqrt(2).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• cons
• nonn

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

From Amiram Eldar, Aug 25 2020: (Start)

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)

Examples

			2.6651441426902251886502972498731398482742113137146594928...
		

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

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• 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.
• R. O. Kuzmin, On a new class of transcendental numbers" (in Russian), Izvestiya Akademii Nauk SSSR, Ser. matem. 7, No. 6 (1930), pp. 585-597.
• Simon Plouffe, 2**sqrt(2), a transcendental number to 5000 digits.
• Simon Plouffe, 2**sqrt(2), a transcendental number to 2000 digits.
• Theodor Schneider, Transzendenzuntersuchungen periodischer Funktionen I. Transzendenz von Potenzen, J. reine angew. Math., Vol. 172 (1935), pp. 65-69.
• Theodor Schneider, Transzendenzuntersuchungen periodischer Funktionen II. Transzendenzeigenschaften elliptischer Funktionen, J. reine angew. Math., Vol. 172 (1934), pp. 70-74.
• Eric Weisstein's World of Mathematics, Gelfond-Schneider Constant.
• Wikipedia, Gelfond-Schneider constant.
• Wikipedia, Gelfond-Schneider theorem.
• Index entries for transcendental numbers

Programs

Mathematica

RealDigits[N[ 2^Sqrt[2], 100]][[1]]

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