Problem - 175
Prove there exist infinite number of positive integer $a$ such that for any positive integer $n$, $n^4 + a$ is not a prime number.
For any integer $m>1$ and positive integer $n$, we have $$n^4 + 4m^4 = (n^2+2mn +2m^2)(n^2-2mn+2m^2)$$
Because $$n^n+2mn+2m^2 > n^2-2mn+2m^2 = (n-m)^2+m^2 \ge m^2 > 1$$, therefore $n^4 +4m^4$ is not a prime number.