18S.Fractions Mod p and Wolstenholme’s Theorem

April 8, 2009 by danielkohen

A23 Prove that if

$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}$

is expressed as a fraction, where $p>3$ is a prime, then $p^2$ divides the numerator.

A24 Let $p>3$ be a prime number and $k=\lfloor\frac{2p}{3}\rfloor$ . Prove that

${p \choose 1}+{p \choose 2}+\cdots+{p \choose k}$

is divisible by $p^2$ .

Here is the official solution file: pen18.pdf

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