OFFSET
1,1
COMMENTS
The other root is the golden ratio phi = 1.618033988... (see A001622).
See also the graph with the two roots (second Mathematica program).
EXAMPLE
3.6015212845193509063188978...
MAPLE
Digits:=200:fsolve(x^sqrt(x+1) - sqrt(x+1)^x=0, x, 3..5);
MATHEMATICA
RealDigits[ FindRoot[x^Sqrt[x+1] == Sqrt[x+1]^x, {x, {3, 5} }, WorkingPrecision -> 105] [[1, 2] ]] [[2]]
(****graph****)
f[x_] := x^Sqrt[x+1] ; g[x_] := Sqrt[x+1]^x
Plot[{f[x], g[x]}, {x, 1, 4}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1, 4}, WorkingPrecision -> 110]
PROG
(PARI) solve(x=3, 4, t=sqrt(x+1); t^x-x^t) \\ Charles R Greathouse IV, Apr 18 2012
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Michel Lagneau, Apr 18 2012
STATUS
approved