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A078333 Decimal expansion of sqrt(2)^sqrt(2). 8
1, 6, 3, 2, 5, 2, 6, 9, 1, 9, 4, 3, 8, 1, 5, 2, 8, 4, 4, 7, 7, 3, 4, 9, 5, 3, 8, 1, 0, 2, 4, 7, 1, 9, 6, 0, 2, 0, 7, 9, 1, 0, 8, 8, 5, 7, 0, 5, 3, 1, 1, 4, 1, 1, 7, 2, 4, 7, 7, 8, 0, 6, 8, 4, 3, 8, 3, 0, 3, 5, 2, 0, 5, 9, 9, 8, 6, 1, 6, 6, 4, 2, 2, 4, 7, 8, 5, 5, 5, 0, 7, 5, 0, 6, 6, 2, 6, 0, 4, 1, 4, 2, 3, 0, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This number was used in a non-constructive proof that an irrational number raised to an irrational power may be a rational number: "sqrt(2)^sqrt(2) is either rational or irrational. If it is rational, our statement is proved. If it is irrational, (sqrt(2)^sqrt(2))^sqrt(2) = 2 proves our statement." (Jarden, 1953). - Amiram Eldar, Aug 14 2020
REFERENCES
Paul R. Halmos, Problems for mathematicians, young and old, The Mathematical Association of America, 1991. Problem 3 B, pp. 22 and 171.
Dov Jarden, Curiosa No. 339: A simple proof that a power of an irrational number to an irrational exponent may be rational, Scripta Mathematica, Vol. 19 (1953), p. 229.
LINKS
J. P. Jones and S. Toporowski, Irrational numbers, American Mathematical Monthly, Vol. 80, No. 4 (1973), pp. 423-424.
Robert Munafo, Notable Properties of Specific Numbers (entry for the number 1.632526919438)
FORMULA
Equals exp(zeta'(1/2, 3) - zeta'(1/2)) = exp((zeta'(-1/2, 3) - zeta'(-1/2))/2), where zeta' is the first derivative of the Hurwitz zeta function and zeta' the first derivative of the Riemann zeta function. - Thomas Scheuerle, Apr 22 2024
EXAMPLE
sqrt(2)^sqrt(2) = 1.632526919438152844773495381...
MATHEMATICA
RealDigits[Sqrt[2]^Sqrt[2], 10, 111][[1]]
PROG
(PARI) 2^.5^.5 \\ Charles R Greathouse IV, Mar 22 2013
CROSSREFS
Cf. A002193.
Square root of A007507. - Michel Marcus, Oct 21 2017
Cf. A185111 (sqrt(2)^sqrt(3)), A185094 (sqrt(3)^sqrt(3)).
Sequence in context: A369876 A010494 A333239 * A302852 A348694 A049605
KEYWORD
nonn,cons
AUTHOR
Robert G. Wilson v, Nov 21 2002
EXTENSIONS
Munafo link clarified by Robert Munafo, Jan 25 2010
STATUS
approved

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Last modified June 19 18:58 EDT 2024. Contains 373507 sequences. (Running on oeis4.)