The Values at the Integers of Power Series that Converge about $z=\infty$

Let the power series in descending powers of $z$

converge outside some circle. Let the numbers $b_1,b_2,\ldots ,b_m$ be rational. If the series assumes integral values for infinitely many integral values of $z$, then $b_0$ is rational and

(Generalization of 93.) [Hurwitz--Courant, p. 30.]

The series

represents an integer for infinitely many integral values of $z$.

Deduce 114 from 188.

Let the power series

that is not everywhere divergent represent integers for all sufficiently large integral values of $z$. Then $F(z)$ is an integral-valued polynomial.

If two polynomials are such that they assume integral values at the same points of the complex plane, then either their sum or their difference is a constant. (This constant is naturally an integer.)

If $m$ is the degree of $y=f(x)$ and $n$ the degree of $z=g(x)$, then we have for sufficiently large $\left|y\right|$

where all $m$ determinations are admissible for $y^{\frac1m}$.