We know that $n!=n(n-1)(n-2)\cdots3\cdot2\cdot1, n\in \mathbb N$. Now I am willing to write $n!$ as $a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$ where $a, r, \alpha\in \mathbb N$.
My question is: how to find such $a$ and $\alpha$? For the second one, I think de Polignac's formula can be use but not sure about it. Can some one help me how to solve this one ? Should I include more criteria on the problems ?
Please help me.