

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
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OFFSET

1,2


COMMENTS

This number was used in a nonconstructive 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. 423424.


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



CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



