A078333 - OEIS

A078333

Decimal expansion of sqrt(2)^sqrt(2).

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

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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

G. C. Greubel, Table of n, a(n) for n = 1..5000

J. Roger Hindley, The Root-2 Proof as an Example of Non-constructivity, 2015.

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)

Wikipedia, Square root of the Gelfond-Schneider constant

Index entries for transcendental numbers

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

Adjacent sequences: A078330 A078331 A078332 * A078334 A078335 A078336