OFFSET
1,1
COMMENTS
The x-th root of x equals the phi-th root of phi: x^(1/x) = phi^(1/phi) = A185261 = 1.3463608200348694434247534661858... .
Not surprisingly, x appears to be irrational. If x is also algebraic, then x^phi would be transcendental by the Gelfond-Schneider theorem.
LINKS
EXAMPLE
6.055722091024741002126639117583149731683828...
x^phi = phi^x = 18.431940924839652158136364051482054378959672... .
MATHEMATICA
{a, b} = NSolve[x^phi == phi^x, x, WorkingPrecision -> 300]; a; RealDigits[N[x/.a, 300]][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Timothy L. Tiffin, Oct 14 2021
STATUS
approved