Determine all possible pair(s) (x, y) of positive integers that satisfy the equation:
x3 = y2 – 15
This problem reminded me of Catalan's Conjecture: prove 8 and 9 are the only consecutive powers. This was proved true in 2002, but for our problem the difference is 15 and not 1.
Catalan's Conjecture has an extension called Pillai's Conjecture which implies that given any difference d, there are a finite number of pairs of powers with a difference of d. So by Pillai's Conjecture there is a finite number of solutions for any difference we choose to put in our problem.