Question

Show that for any integer n, the number $n^4-20n^2+4$ is not a prime number.

Answer 1

We can factorize $n^4-20n^2+4=(n^2-4n-2)(n^2+4n-2)$

For this to be a prime number, exactly one of the factors must be equal to $1$

Now,

$n^2-4n-2=(n-2{)}^2-6$ and

$n^2+4n-2=(n+2{)}^2-6$

For any of these factors to be $1$ , $n$ will not be an integer

So, $n^4-20n^2+4$ is not prime for any value of $n$