@ErickWong Brocard's problem implies $ \log(a)-\log(r)=n- s_2(n)-2 $ where $n!=2^{n- s_2(n)}\times a \times r$ ($a, r$ are odd numbers). Isn't it sufficient to show that Brocard's problem has finite solutions if $ \log(a)-\log(r) > n$ (induction), $\forall n>n_0$, for sufficient large $n_0$?

Probably, showing $ \log(a)-\log(r) \neq n$ by induction, if possible.

I mean, $ \log(a)-\log(r) \neq n- s_2(n)-2 $

@ErickWong , u r correct about $a, r$ and , yes, $\log a - \log r > n$ or $\log a - \log r < n-s_2(n)-2$ will not uniformly hold over all choices/all possible decompositions of $a$ and $r$, but it seems quite possible to show that either $\log a - \log r < n-s_2(n)-2$ or $\log a - \log r > n-s_2(n)-2$ , in other words, to show $ \log(a)-\log(r) \neq n- s_2(n)-2 $, $\forall n>n_0$, for sufficient large $n_0$ , I am considering the possibility.