Exponential Diophantine representation of the factorial

How can I use the identity $$\sum_{n=0}^\infty \frac{\tau^n}{n!} = \lim_{y \to \infty} \left( 1 + \frac{\tau}{y} \right)^y$$ to find an exponential Diophantine representation of the factorial?

I was hoping to use a very large value of $y$ (so that the RHS is within an integer of the actual number) and expand with binomial theorem then extract the $\tau^k$ term to get at $k!$ but since we have $\frac{1}{k!}$ this is impossible.

I am really stuck! Any clues on how to do this? Thank you.

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