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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?

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marty cohen
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    Maybe I'm missing something, but don't they use a property of $e$ in the very first section, where they perform the integration by parts? – Alex Wertheim Aug 07 '13 at 18:55
  • I think I see what you mean. I was just interpreting that as a property of $e^x$, not a property of $e$. I will meditate on this a bit. Thanks. – marty cohen Aug 07 '13 at 18:59
  • Where does he define what $f$ is? – Steven Alexis Gregory May 01 '15 at 22:24
  • Initially, $f$ is unspecified. Then this: "If f is a polynomial of degree m". In equation (4), $f$ is specified as a polynomial which depends on a large prime $p$ and $d$, the degree of the polynomial that $e$ is assumed to satisfy. – marty cohen May 02 '15 at 00:27
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    The link to the proof appears no long appears to be valid (bad URL). Can you present the proof here or supply an alternative link? – Mark Yasuda May 31 '16 at 04:04

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