5th Swedish 1965 Problem 2 Find all positive integers m, n such that m 3 - n 3 = 999.
10 3 - 1 3 , 12 3 - 9 3 Solution The fastest way to do this is simply to write down the small cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197 (those at least you should know), 2744, 3375, 4096, 4913, 5832, 6859. Note that 6859 - 5832 > 999 and (n+1) 3 - n 3 is obviously an increasing function of n, so there are no solutions for m ≥ 19. We find the solutions for m < 19 by inspection. 5th Swedish 1965 © John Scholes jscholes@kalva.demon.co.uk 30 December 2003 Last corrected/updated 30 Dec 03