# Wolstenholme's Theorem, Vandermonde's Convolution Identity and so much more

Number Theory Level 5

Let $$p=2^{16}+1$$ be an odd prime. Define $$\large{H_{n}=\sum _{ x=1 }^{ n }{ \frac { 1 }{ n } } }$$. Find the remainder when

$\Large{\left( p-1 \right) !\sum _{ n=1 }^{ p-1 }{ { H }_{ n } } \cdot { 4 }^{ n }\cdot \left( \begin{matrix} 2p-2n \\ p-n \end{matrix} \right) }$

is divided by $$p$$

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