Determine all possible pair(s) (x, y) of positive integers that satisfy the equation:
x^{3} = y^{2} – 15

(In reply to

re(2): Fascinating by broll)

I agree we have not proven any limitation on possible solutions to the original or variants. I have tested all the odd subtrahends up thru 51. We've already reviewed (x=109,y=1138) for the original, and I noted (x=5234,y=378661) for "- 17".

The only other case I found (other than obvious single-digit solutions for subtractions of 1, 3, 37, and 41) of interest was for (x=243,Y=3788) if subtracting 37. All of my testing was limited by the number of significant digits I could work with, testing each equation for 0 < x < 10000. (1259712**2 = 11664**3 without subtracting anything on the right side). The only way to disprove a conjecture would seem to be some construction related to factorization. We at least know that higher solutions would be very sparse, if existing at all. An interesting problem -- hope KS has a solution for us.