Diophantine Approximations and a Problem from the 1988 IMO
Charles Osgood2006, Rocky Mountain Journal of Mathematics
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Abstract
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Harborth has shown how to describe all integer solutions to a Diophantine equation arising from a problem at the 1988 International Mathematical Olympiad using a clever reduction method. This paper extends Harborth's results through the classical theory of continued fractions by addressing shortcomings in existing methodologies, particularly in Legendre's theorem. The paper provides necessary conditions for rational integers in relation to continued fractions and applies these findings to facilitate the resolution of certain quadratic Diophantine equations.
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