1. The problem statement, all variables and given/known data Find all the primes p and q such that [itex]p^2-p-1=q^3[/itex] 3. The attempt at a solution Trying out the first prime number 2, it is clear that [itex]p>q[/itex] and that the difference is larger than 1. Then when factorizing it [itex]p^2-p=q^3+1[/itex] [itex]\Rightarrow[/itex] [itex]p(p-1)=(q+1)(q^2-q+1)[/itex], I get that p must divide [itex]q^2-q+1[/itex], since p is larger than q+1. However, these are all the observations I have made and I don't have a good idea how to continue. Hints are welcomed. With some testing with my computer, I found a solution with 11 and 37 and no others in near sight.