There are infinitely many prime numbers expressible in the form $n^3+1$ where $n$ is a positive integer.
I am not sure whether this is true or not. I tried to prove that it is true via contradition:
Assume via contradiction that there are finite number of prime numbers that can be written in the form $n^3+1$ where $n$ is a positive integer. Then there exists an $N$ such that $N^3+1=P_N$ , where $P_N$ is a prime number and for all $n\ge N$, $n^3+1$ is composite. My idea was to create another prime number greater than $N$ that can be written in this form potentially $(2N)^3+1$ to produce a contradiction. But I am not sure how to construct such a prime number.Can anyone provide hints as to how to approach this problem? I would appreciate hints more than answers.