Solving Diophantine Equations using Bronze Ratio

Abstract

The study of Diophantine equations has been of great interest among mathematicians for several centuries. In this paper, we will introduce a special type of pair of Diophantine Equations and will solve them using the continued fraction expansion of bronze ratio. In particular, the successive convergents of the continued fraction discussed in this paper provide the complete solutions to the given equations. Incidentally the solutions depend on one of the most important real numbers called Bronze Ratio. The method adopted to solve the given equations is novel and provide insights to solve many more similar equations.