%I #39 Aug 21 2023 12:44:19
%S 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,
%T 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,
%U 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
%N Decimal expansion of sqrt(2)^sqrt(2).
%C 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
%D Paul R. Halmos, Problems for mathematicians, young and old, The Mathematical Association of America, 1991. Problem 3 B, pp. 22 and 171.
%D 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.
%H G. C. Greubel, <a href="/A078333/b078333.txt">Table of n, a(n) for n = 1..5000</a>
%H J. Roger Hindley, <a href="http://www.users.waitrose.com/~hindley/Root2Proof2015.pdf">The Root-2 Proof as an Example of Non-constructivity</a>, 2015.
%H J. P. Jones and S. Toporowski, <a href="http://www.jstor.org/stable/2319091">Irrational numbers</a>, American Mathematical Monthly, Vol. 80, No. 4 (1973), pp. 423-424.
%H Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers-2.html">Notable Properties of Specific Numbers</a> (entry for the number 1.632526919438)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant#Properties">Square root of the Gelfond-Schneider constant</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e sqrt(2)^sqrt(2) = 1.632526919438152844773495381...
%t RealDigits[Sqrt[2]^Sqrt[2], 10, 111][[1]]
%o (PARI) 2^.5^.5 \\ _Charles R Greathouse IV_, Mar 22 2013
%Y Cf. A002193.
%Y Square root of A007507. - _Michel Marcus_, Oct 21 2017
%Y Cf. A185111 (sqrt(2)^sqrt(3)), A185094 (sqrt(3)^sqrt(3)).
%K nonn,cons
%O 1,2
%A _Robert G. Wilson v_, Nov 21 2002
%E Munafo link clarified by _Robert Munafo_, Jan 25 2010
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