IMO 1982 SL 16

Problem

C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the equation x3 −3xy2 + y3 = n has a solution in integers (x, y), then it has at least three such solutions. Show that the equation has no solution in integers when n = 2891.

Solution

It is easy to verify that whenever (x, y) is a solution of the equation x3 −3xy2 + y3 = n, so are the pairs (y −x, −x) and (−y, x −y). No two of these three solutions are equal unless x = y = n = 0. Observe that 2981 \equiv2 (mod 9). Since x3, y3 \equiv0, \pm1 (mod 9), x3 − 3xy2 + y3 cannot give the remainder 2 when divided by 9. Hence the above equation for n = 2981 has no integer solutions.