I think that value means the result obtained by placing a real number in place of x, e.g. the f(x).
I translate them using Google, trying the result to make sense.
In Russian the exact wording was
Quote:
Существует ли квадратный трехчлен с целыми коэффициентами, все значения которого в натуральных точках — натуральные степени двойки?
Let the trinomial be $f(x) = ax^2+bx + 2^kc$ where $k$ is a nonnegative integer and $c$ is odd. Then, $f(2^{k+1}) = 2^{k+1}(someinteger)+ 2^kc$ which can't be a power of 2. We have a counterexample; hence, there is no quadratic trinomial that exists.