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.

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