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

Offset: 1

Author

Robert G. Wilson v, Nov 21 2002

Keywords

• nonn
• cons

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

Examples

			sqrt(2)^sqrt(2) = 1.632526919438152844773495381...
		

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

Crossrefs

Cf. A002193.

Square root of A007507. - Michel Marcus, Oct 21 2017

Cf. A185111 (sqrt(2)^sqrt(3)), A185094 (sqrt(3)^sqrt(3)).

Programs

Mathematica

RealDigits[Sqrt[2]^Sqrt[2], 10, 111][[1]]

PARI

2^.5^.5 \\ Charles R Greathouse IV, Mar 22 2013

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

Extensions

Munafo link clarified by Robert Munafo, Jan 25 2010