(p-1)! number theory problem

Question

I was working on this little problem:

Let $\frac{a}{(p-1)!} = 1 + 1/2 + 1/3 + ... + 1/(p-1)$, where p is a positive prime.

(a) Prove $p\mid a$. (b) Can $p^2 \mid a$?

I thought (a) was pretty self-explanatory with Wilson's theorem. For (b), it seems if $p\geq 5$ that it holds (I checked for $p= 5, 7, 11, 13, 17$). I tried to work with $\mod p^2$ but got stuck. Any hints?

Thanks.

Answer 1

$(a):$ Observe that $$\sum_{r=1}^{p-1}\frac1r\equiv\sum_{r=1}^{p-1}r\pmod p$$

Now, $\displaystyle\sum_{r=1}^{p-1}r=\frac{p(p-1)}2\equiv0\pmod p$ for prime $p\ge3$