Let $n$ be an integer. Then any prime factor of
$$ 5 n^4 - 70 n^3 + 380 n^2 - 945 n + 911 $$
Must be congruent to 1 mod 10.
Also
Let $n$ be an integer. Then any prime factor of
$$ 5 n^4 - 10 n^3 + 20 n^2 - 15 n + 11 $$
Must be congruent to 1 mod 10.
How to prove these ?
How to find such identities ?
Hint: By factoring the first polynomial minus one we get, by setting $p(n)=5 n^4 - 70 n^3 + 380 n^2 - 945 n + 911$,
$$ 4\cdot p(n) = -1+5\left(2n^2-14n+27\right)^2 $$ so for every odd prime divisor $q\mid p(n)$ we have $\left(\frac{5}{q}\right)=+1$ and by quadratic reciprocity $q$ is forced to belong to some particular residue classes $\!\!\pmod{20}$. It is not difficult to show that $p(n)$ is odd for any $n\in\mathbb{Z}$ and similar arguments apply to the second polynomial, too.