Let
![P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}](/media/m/1/a/1/1a18f92c372592c84e57150dae870515.png)
, where
![a_{0},\ldots,a_{n}](/media/m/4/d/1/4d17221d71ad54e599b7ab5cb60445a4.png)
are integers,
![a_{n}>0](/media/m/5/1/5/515b2978792cfcaa0056deb658f9623e.png)
,
![n\geq 2](/media/m/e/d/b/edbb3c15913fef4235c90cca2333a608.png)
. Prove that there exists a positive integer
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
such that
![P(m!)](/media/m/f/e/e/fee97c0dbf70943d5575829aacb97a0a.png)
is a composite number.
%V0
Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.