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Harborth has recently shown how to describe all integer solutions to a Diophantine equation arising from a problem at the 1988 International Mathematical Olympiad. Harborth uses a clever reduction method, although it seems that this method is somewhat ad hoc. The purpose of the present paper is to show how the result of Harborth can be proved, and extended, using the classical theory of continued fractions. More to the point, a shortcoming in this classical theory is circumvented by an extension to Legendre's theorem concerning a sufficient condition for a rational integer to be a convergent to a given irrational number.

Diophantine Approximations and a Problem from the 1988 IMO