A360148 Decimal expansion of the nontrivial number x for which x^sqrt(2) = sqrt(2)^x.

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

Offset: 1

Author

Timothy L. Tiffin, Jan 27 2023

Keywords

• nonn
• cons

Comments

Not surprisingly, x appears to be irrational. If x is also algebraic, then x^sqrt(2) would be transcendental by the Gelfond-Schneider theorem.

x is irrational by the Lindemann-Weierstrass theorem. - Charles R Greathouse IV, Jan 27 2023

x = W(-1,-log(2)/(2*sqrt(2)))*-2*sqrt(2)/log(2) = e^-W(-1,-log(2)/(2*sqrt(2))), where W(-1,z) is branch -1 of the Lambert W function. (Branch 0 returns sqrt(2).) Together with sqrt(2), x is unique over the complex numbers as well as the reals. - Natalia L. Skirrow, Jun 22 2023

Examples

			8.937437066059062316820208064624691048717068...
		

Programs

Mathematica

{a, b} = NSolve[x^Sqrt[2] == Sqrt[2]^x, x,
  WorkingPrecision -> 300]; a; RealDigits[N[x /. b, 300]][[1]]
N[LambertW[-1,-Log[2]/(2*Sqrt[2])]*-2*Sqrt[2]/Log[2],300] (* Natalia L. Skirrow, Jun 22 2023 *)

Formula

From Natalia L. Skirrow, Jun 22 2023: (Start)

Newton's method gives x' = x - (x^sqrt(2) - sqrt(2)^x)/(sqrt(2)*x^(sqrt(2)-1) - sqrt(2)^x*log(2)/2).

Taking logs first gives x' = x - (sqrt(2)*log(x) - x*log(2)/2)/(sqrt(2)/x - log(2)/2).

Beginning with x^(2/x)=sqrt(2)^sqrt(2) instead gives x' = x - (2^(1/sqrt(2)) - x^(2/x))/(log(x) - 1).

(End)