Question If p > 3 is a prime and $ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p} = \frac{a}{b}$ then prove that $p^{4} \mid (ap-b)$.
There is an exercise in Herstein which states, if p > 3 is prime and if $1 + \frac{1}{2} + \cdots + \frac{1}{p-1} = \frac{a}{b}$ then $p^{2} \mid a$. (Page 116, Problem 2). But i am not sure whether this helps or not. Hints, suggestions would be of great help!