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Let $p$ be an odd prime and let $1 + \frac{1}{2} + \cdots + \frac 1{p-1} = \frac ab$, where $a,b$ are integers. Show that $p\mid a$. (Hint: As $a$ runs through $U_p$, so does $a^{-1}$.)

P.S. $U_p$ means the group of of invertible elements in $\mathbb{Z}_p$, and I believe for prime $p$, $U_p = \mathbb{Z}_p$.

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    $\begingroup$ For prime $p$, $U_p = \mathbb{Z}_p \setminus \{0\}$. $\endgroup$
    – Kaj Hansen
    Commented Mar 24, 2015 at 6:27

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Hint: If you take $\pmod p$ on both side, do you see that $\sum_\limits {n=1}^{p-1} 1/n \equiv \sum \limits_ {n=1} ^{p-1} n \pmod p $ ?

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  • $\begingroup$ Oh, this is what "run through" mean. Thank you so much! $\endgroup$
    – MonkeyKing
    Commented Mar 24, 2015 at 7:14

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