I encountered some other problem and I found a beautiful proof here Write $1/1 + 1/2 + …1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$. (see Thomas Andrew's post)
But I thought he may miss the proof that $(a_1,(p-1)!)=1,$ which is not trivial for me. (As for the definitions of $a_1$ and $p$ please see the link above. The definitions are simple and clear there.)
My problem is just that how to prove $(a_1,(p-1)!)=1$.
I tried to use the property that $(n,m)=(n.n+km)$ for any integers $n,m,k.$ But it turns out it makes the expression messy and dirty. And I can't go further.
Any help will be thanked.
Best Answer
By definition $$a_1=\sum_{n=1}^{p-1}(\prod_{m\ne n}m).$$ So in fact $a_1$ and $(p-1)!$ need not be co-prime (for $p=7$, we have $a_1=1764$ and $(p-1)!$ is even). The proof in the linked question only requires that $p$ does not divide $(p-1)!$, which is clear: $(p-1)!$ is divisible only by primes $<p$.
P.S. In case you are wondering why the proof only requires that $p$ does not divide $(p-1)!$, notice that we can reduce the fraction $\frac{a_1}{(p-1)!}$ to the lowest terms $\frac ab$, where $a=\frac{a_1}{\gcd(a_1,(p-1)!)}$. Since $p$ does not divide $(p-1)!$, $\frac{a_1}{\gcd(a_1,(p-1)!)}$is divisible by $p^2$, hence concluding the proof.
Hope this helps.