I discovered this site which claims that “7 is the only prime followed by a cube”. I find this statement rather surprising. Is this true? Where might I find a proof that shows this?
In my searching, I found this question, which is similar but the answers seem focused on squares next to cubes.
This is certainly true. Suppose n3−1 is prime, for some n. We get that n3−1=(n−1)(n2+n+1) and so we have that n−1 divides n3−1. If n−1>1 then we’re done, as we have a contradiction to n3−1 being prime.