Proof of Irrationality of e using Diophantine Equations I was trying to prove that e is irrational without using the typical series expansion, so starting off  $e = a/b $  Take the natural log so  $1 = \ln(a/b)$ Then $1 = \ln(a)-\ln(b)$ So unless I did something horribly wrong showing the irrationality of $e$ is the same as showing that the equation $c = \ln(a)-\ln(b)$ or $1 = \ln(a) - \ln(b)$ (whichever one is easiest) has no solutions amongst the natural numbers. I feel like this would probably be easiest with infinite descent, but I'm in high school so my understanding of infinite descent is pretty hazy. If any of you can provide a proof of that, that would be awesome.
EDIT: What I mean by "typical series expansion" is Fourier's proof http://en.wikipedia.org/wiki/Proof_that_e_is_irrational#Proof

One situation in which the existence of a solution 
to a Diophantine equation implies an irrationality result is this:
If, for a positive integer $n$, there are
positive integers $x$ and $y$ satisfying
$x^2 - n y^2 = 1$,
then $\sqrt n$ is irrational.
I find this amusing, since this proof
is more complicated than any of the standard proofs that
$\sqrt n$ is irrational and also, since it does not
assume the existence of solutions to
$x^2 - n y^2 = 1$,
requires the "user"
to supply a solution.
Solutions are readily supplied for 2 and 5,
but are harder to find for 61.
