14th ASU 1980 Problem 5 Are there any solutions in positive integers to a 4 = b 3 + c 2 ?
We have b 3 = (a 2 - c)(a 2 + c), so one possibility is that a2 ± c are both cubes. So we want two cubes whose sum is twice a square. Looking at the small cubes, we soon find 8 + 64 = 2·36 giving 6 4 = 28 2 + 8 3 . Multiplying through by k 12 gives an infinite family of solutions. Note that the question does not ask for all solutions. 14th ASU 1980 © John Scholes jscholes@kalva.demon.co.uk 4 December 2003 Last corrected/updated 4 Dec 03