This is a common problem.
Try searching the internet
Let be the th harmonic number. I started with induction and easily arrived at is not an integer so assume that where . Then .
But I realize now that this is a dead end. How do I proceed? I know I am trying to show that the denominator of the sum is even while the numerator is odd.
While the internet is useful, the site you have linked seems to be a collection of rough sketches of proofs; which I had a bit of trouble fleshing out. When I tried to follow similar arguments to construct a proof, I found that things got very messy.
I did several internet searches before I made my post, but all the arguments I found seemed either: ad-hoc, or dependent on p-adic arguments. And I don't know what p-adic is so...
I have constructed the following proof. I didn't see anything like this in the online proofs that I found, so I'm worried that my proof is incomplete.
If D={1,2}, then m- ½=1=an integer. It’s a trivial case.
m = r/s + 1/p
=>(pr+s)/ps=m
=>s = mps-pr
=>s=p(ms-r)
i.e. p|s which is a contradiction due to the defn of s. This might suffice the proof. Your approach was excellent.