OFFSET
1,1
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. - Nathan L. Skirrow, Jun 22 2023
FORMULA
From Nathan 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)
EXAMPLE
8.937437066059062316820208064624691048717068...
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] (* Nathan L. Skirrow, Jun 22 2023 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Timothy L. Tiffin, Jan 27 2023
STATUS
approved