If $(3+ \sqrt2 )^n +r^n$ is always integer, then there exists $r=3-\sqrt2$ satisfying the equation.
Then, given $(1+\sqrt2 +\sqrt3 +\sqrt6 )^n +x_1^n+x_2^n +...+x_m^n$ is an integer for all natural numbers $n$, the smallest natural number $m$ such that there exists $(x_1,...,x_m)$ is $3$?
And, $x_1=1+\sqrt2 -\sqrt3 -\sqrt6, x_2=1-\sqrt2 +\sqrt3 -\sqrt6, x_3=1-\sqrt2 -\sqrt3 +\sqrt6$ ?