Let $ p$ be an odd prime number which large enough. I am interested in the study of the sums of reciprocals in the field $ \mathbb{F}_p$ . In particular, I have the following question: which primes $ p$ do satisfy \begin{equation*} \sum_{j\equiv 1\pmod{5}}^{\frac{p-1}{2}}j^{-1}\not\equiv\sum_{j\equiv 3\pmod{5}}^{\frac{p-1}{2}}j^{-1}\pmod{p}\quad ? \end{equation*} Thanks for any comments or any helpful references.Read more