Let $n$ be a positive integer.
Define $$f(n)=\prod_{j=1}^n (x-j)+\prod_{j=1}^n (x-(j+n))$$
If $n>1$ is odd, then it is easy to show $$(2x-2n-1)\mid f(n)$$
My conjecture is the following :
- For even $n$ , $\frac{f(n)}{2}\in \mathbb Z[x]$
- For even $n$ , $\frac{f(n)}{2}$ is irreducible over $\mathbb Z$
- For odd $n>1$ , $\frac{f(n)}{2x-2n-1}$ is irreducible over $\mathbb Z$
The conjecture is true upto $n=200$. Can we prove it ?