f entire $f(n)=n^5$ and $f (-\frac{n}{2} )=n^7$ . How many zeroes have $(f-\pi)(f-e)$ without using picard

Suppose f is an entire function satisfying $f(n)=n^5$ and $f (-\frac{n}{2} )=n^7 \forall n\in \Bbb Z_{>0}$. How many zeros does the function $g(z)= (f(z)-e)(g(z)-\pi)$ have?

Well... First note that $\infty$ is an essential singularity. So by Picard Theorem, every neighborhood of $\infty$ is surjective on $\Bbb C$ except at most for one point, and every other point have infinite preimages.

The problem is that I can't use Picard theorem, because we have not proved it. There is a way to do this only with elementary facts?

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Is $g$ the same as $f$? – user53153 Dec 19 '12 at 4:46