# Prove or Disprove the following Conjecture

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.

• Hint: $n^3+1 = (n+1)(n^2-n+1)$. – Robert Israel Mar 30 '20 at 22:19

## 1 Answer

Hint:

$$n^3 + 1 = (n+1)(n^2-n+1)$$

• If I am understanding correctly $n^3+1=(n+1)(n^2-n+1)$ since this is prime then either $n+1=1$ or $(n^2-n+1)=1$. Note $n+1\neq 1$ since this implies $n=0$ not a positive number and $n^3+1=1$ not a prime so $(n^2-n+1)=1$ which implies $(n-1)=0$ resulting $n=1$ so the only prime that can be written in this form is 2. So the conjecture is false. – Noe Vidales Mar 30 '20 at 22:26
• Great stuff, indeed! – Victor Mar 30 '20 at 22:29