## Three random conjectures

Please refute at least one of the following:

$\displaystyle \sum_{i=1}^n \dfrac{1}{i} \equiv 0 \pmod{p^2}$ implies $n < p^2$

$\displaystyle \sum_{i=1}^n \dfrac{1}{i} \equiv 0 \pmod{p^3}$ implies $n = p - 1$

$\displaystyle \sum_{i=1}^n \dfrac{1}{i} \equiv 0 \pmod{p^4}$ never happens

EDIT. $p = 11$, $n = 10583$ is quite a cute counterexample to the first $2$ of these.