## Another random conjecture

In spirit of yesterdays post, could the following be true, wild as it is?;

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

If true, $n_p = O(p^p)$ can’t be far from the truth. Do we have $n_p = O(p^k)$? I very much doubt it.. $n_3 = 22$, $n_5 = 24$ and $n_7 \ge 16735 \approx 7^5$ in any case.

EDIT: By just googling “harmonic number 16735” I found some articles from around 1990, that are devoted to exactly this conjecture and it seems like it’s true (it’s proven for all $p < 550$ with the possible exception of $83$, $127$ and $397$), but hard. The only remaining conjecture from the previous post seems to hold up too. Anyway, they use p-adic methods, which I am (still) not that familiar with, so I’m definitely going to search for some literature about it.