%I #10 May 14 2019 21:40:25

%S 6,3,6,2,6,2,9,3,9,2,9,4,5,3,1,0,1,9,9,8,7,5,1,3,7,5,5,2,0,4,2,3,3,1,

%T 7,3,1,1,7,8,6,7,0,5,7,9,3,6,2,6,2,2,9,4,8,8,6,5,4,0,6,4,5,4,0,6,3,8,

%U 9,2,1,4,4,0,2,7,9,9,2,7,3,3,9,0,9,1,4,8,0,5,4,8,9,4,6,9,6,2,0,7

%N Decimal expansion of the larger solution to x^x = 3/4.

%C Since (1/e)^(1/e) < 3/4 < 1, the equation x^x = 3/4 has two solutions x = a and x = b with 0 < a < 1/e < b < 1. Both solutions are transcendental (see Proposition 2.2 in Sondow-Marques 2010).

%H J. Sondow and D. Marques, <a href="http://arxiv.org/abs/1108.6096">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae 37 (2010) 151-164.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.636262939294531019987513755204233173117867057936262294886540645406389214402799...

%t x = x /. FindRoot[x^x == 3/4, {x, 0.7}, WorkingPrecision -> 120]; RealDigits[x, 10, 100] // First

%Y Cf. A030798 (x^x = 2), A072364 ((1/e)^(1/e)), A194624 (smaller solution to x^x = 3/4).

%K nonn,cons

%O 0,1

%A _Jonathan Sondow_, Sep 02 2011