Determine whether there exists a prime $q$ so that for any prime $p$ the number $$\sqrt[3]{p^2+q}$$is never an integer.
Source: 2018 Latvia BW TST P13
Tags: number theory, number theory unsolved, prime numbers
Determine whether there exists a prime $q$ so that for any prime $p$ the number $$\sqrt[3]{p^2+q}$$is never an integer.