There is this problem that I tried but there are still some questions, confusions and doubts.
There exist infinitely many natural numbers $a$, such that for any natural number $n$, the number $z=n^4 + a$ is a composite number.
How I did it: Let $a=3n^4$, so $z=n^4+3n^4=(2n^2)^2$, since $2n^2>1$, so $z$ is composite.
First I am not sure whether my approach is correct, because for any $a$, it is not the case that there is any natural number $n$ so that $a=3n^4$. Try $a=1$, then $1=3n^4$, then $n$ cannot be a natural number.
Is there something wrong with my solution? Or is there something wrong with my doubt? If so, how can we then solve the problem?
Many many thanks!