A348359 Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.

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, 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, 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

Offset: 1

Author

Timothy L. Tiffin, Oct 14 2021

Keywords

• nonn
• cons

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.

Examples

			6.055722091024741002126639117583149731683828...
x^phi = phi^x  = 18.431940924839652158136364051482054378959672... .
		

Links

• Wikipedia, Gelfond-Schneider theorem

Crossrefs

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

Programs

Mathematica

RealDigits[x/.FindRoot[x^GoldenRatio==GoldenRatio^x,{x,6},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Dec 09 2024 *)

Extensions

Prior Mathematica program replaced by Harvey P. Dale, Dec 09 2024