I have solved the following three problems of my book:
Prove that the $n$th harmonic number $H_n = \sum\limits^n_{k=1} \dfrac{1}{k}$ is never an integer for $n \ge 2.$
Prove that none of the $2^{n+1}$ numbers $\pm \frac{1}{k} \pm \frac{1}{k+1} \pm \cdots \pm \frac{1}{k+n}$, where we consider all combinations of plus and minus signs, is an integer, for any positive integers $n$ and $k$.
Prove that $\sum\limits^n_{k=1} \dfrac{1}{2k-1}$ is never an integer for $n>1.$
These problems can be solved by at least two different ways: One, for example on the first problem, considering the highest power of 2 on the set $\{1, 2, \ldots, n\}$ and then checking divisibility by 2 supposing it is an integers, and the second method is to prove by induction that there is integers $j, \ell$ such that $H_n = \frac{2j+1}{2\ell}.$ With these ideas I've got this generalization:
If $p$ is a prime number, then $\sum\limits^n_{k=1} \dfrac{1}{(p-1)k - (p-2)}$ is never an integer for $n>1.$
Then I considered this more generic problem:
Problem. Show that $\sum\limits^n_{j=1} \dfrac{1}{k + mj}$ is never an integer for positive integers $n, m, k.$
But on this problem, the two later ideas doesn't work in general. I've found in "mathworld.wolfram" website that Erdős proved it on 193X or something. How?