uva1434

\[\begin{align*} S_n=\sum_{k=1}^{n}\left[\frac{(3k+6)!+1}{3k+7} -\left[ \frac{(3k+6)!}{3k+7}\right]\right] \end{align*} \]

威尔逊定理:\((p-1)!\equiv-1(mod\ p)\)其中p为质数

若对于某个k,有3k+7为质数,则

设\(p=3k+7\)

\[\frac{(3k+6)!+1}{3k+7} -\left[ \frac{(3k+6)!}{3k+7}\right]=\frac{(p-1)!+1}{p}-\left[ \frac{(p-1)!}{p}\right]\\ \because (p-1)!\equiv-1(mod\ p)\\ \therefore (p-1)!=ap-1\\\frac{(p-1)!+1}{p}-\left[ \frac{(p-1)!}{p}\right]=\frac{ap-1+1}{p}-\left[\frac{ap-1+1}{p}\right]=a-a+1=1\\\nonumber \]

3k+7不为质数,则为0

详细的还不会证...一会看看题解