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Decimal expansion of the nontrivial number x for which x^sqrt(2) = sqrt(2)^x.
1

%I #19 Jul 03 2023 01:05:03

%S 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,

%T 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,

%U 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

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

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

%C x is irrational by the Lindemann-Weierstrass theorem. - _Charles R Greathouse IV_, Jan 27 2023

%C 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

%F From _Nathan L. Skirrow_, Jun 22 2023: (Start)

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

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

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

%F (End)

%e 8.937437066059062316820208064624691048717068...

%t {a, b} = NSolve[x^Sqrt[2] == Sqrt[2]^x, x,

%t WorkingPrecision -> 300]; a; RealDigits[N[x /. b, 300]][[1]]

%t N[LambertW[-1,-Log[2]/(2*Sqrt[2])]*-2*Sqrt[2]/Log[2],300] (* _Nathan L. Skirrow_, Jun 22 2023 *)

%K nonn,cons

%O 1,1

%A _Timothy L. Tiffin_, Jan 27 2023