A348359 - OEIS

%I #14 Feb 13 2022 02:55:06

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

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

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

%N Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.

%C The x-th root of x equals the phi-th root of phi: x^(1/x) = phi^(1/phi) = A185261 = 1.3463608200348694434247534661858... .

%C Not surprisingly, x appears to be irrational. If x is also algebraic, then x^phi would be transcendental by the Gelfond-Schneider theorem.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem">Gelfond-Schneider theorem</a>

%e 6.055722091024741002126639117583149731683828...

%e x^phi = phi^x  = 18.431940924839652158136364051482054378959672... .

%t {a, b} = NSolve[x^phi == phi^x, x, WorkingPrecision -> 300]; a; RealDigits[N[x/.a, 300]][[1]]

%Y Cf. A001622 (phi), A094214 (1/phi), A185261 (phi^(1/phi)), A073226 (e^e, see first comment).

%K nonn,cons

%O 1,1

%A _Timothy L. Tiffin_, Oct 14 2021