Problem 5 of the Canadian Mathematical Olympiad 1972

Prove that the equation x +11 = y has no solution in positive integers x and y.

Solution by Steve Dinh, a.k.a. Vo Duc Dien

Rearrange the equation to (x y) + 3xy 3xy = -11 or

(y x) [ (y x) + 3xy ] = 11

Since 11 is a prime integer, the only possible values for y x are 1, 11, 11 or 11

If y x = 1 then 3xy = 11 -1 = 1330 or xy = 1330/3 which is not an integer and therefore this is not a possible scenario.

If y x = 11 then 3xy = 0 and either x or y must be 0 and not positive as required.

If y x = 11 then 3xy = 11 - 11 or xy <0 and either x or y must be negative and not both being positive as required.

If y x = 11 then 3xy = 1 - 11 or xy <0 which is the same as the previous case.