A number is prime if it is divisible by itself and 1.

We have been given, n³ + 1 to be, there exist only one value.

n³ + 1

On factorising,

= n³ + 1³

Using basic indentity a³ + b³ = (a+b)(a² - ab + b²)

or,  n³ + 1³ = (n + 1)(n² − n + 1)

for n to be prime, either (n + 1) = 1 or (n² - n + 1) = 1

n + 1 = 1 is not possible, ∀n∈N

or, (n² - n + 1) = 1

or, n² - n + 1 - 1 = 0

or, n(n-1) = 0

n = 0  or n - 1 = 0 i.e n = 1

neglecting n=0 as 1 is not a prime.

if n = 1,

n³ + 1 = 1 + 1 = 2, which is obviously a prime.

Therefore, in the form, n³+1, the only prime number that exists is 2.

Answer:

It can be prime only when n=30

And,9001 is a prime number.