This started with my question "Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?". Kunnysan suggested that I model a proof on the standard proof that $e$ is transcendental.
I searched for a proof and found this one: http://rutherglen.science.mq.edu.au/math334s106/m2334.Dioph.e.pdf.
Reading this proof, I can understand almpst all of it, but I have one big problem: I do not see where any property of $e$ is actually used.
I see where they assume $\sum^d_{k=0} a_ke^k = 0 $. They then show that, for large enough primes $p$, a contradiction ensues.
But I do not see where any property of $e$ (such as $e = \sum_{n=0}^{\infty} 1/n!$ or $e = \exp(1)$ and $\exp(x) = \exp'(x)$) is used. It almost looks like this could be used to prove any number is transcendental.
I feel dense.
So, my question is, where in the proof is a property of $e$ used?