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Let $V_p$ be the $p$-adic valuation. We know that $(p - 1)! + 1\equiv0\mod p$ for the prime $p$ by Wilson's theorem. I wonder if there is an upper bound for $V_p((p - 1)! + 1)$.

Also I do not know how to prove the following statement: Let $p\equiv 7\mod8$ be a prime. Then $\sum\limits_{r = 1}^{\frac{p - 1}{2}}r(\frac{r}{p}) = 0$, where $(\frac{\cdot}{\cdot})$ is the Legendre symbol.

Could anybody help me to answer these questions? Thanks.

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    @GerryMyerson: The lower bound in the OEIS entry has meanwhile been updated to $2\cdot10^{13}$.2016-06-28

1 Answers 1

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$\lfloor \frac{n-1}{p-1} \rfloor - k \leq v_p(n !) \leq \lfloor \frac{n-1}{p-1} \rfloor$

with $k$ defined by $ p^k \leq n < p^{k+1} $

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    You can make $\lfloor$ and $\rfloor$ (or any pair of delimiters) adapt to the size of their content by preceding them with `\left` and `\right`, respectively.2016-06-28