A new relation to find all natural solutions to hyperbola and Diophantine equations using Pell's equation

Abstract EN

In this paper a new relation has been found between any natural solution to the hyperbola equationx^2-dy^2=c^2, where cϵN,d a positive square free numberand a form that gives allnatural solutions to Pell'sequationx^2-dy^2=1.

Which enable us to obtain all natural solutions for the above hyperbola equation thena square natural numberu for the Diophantine equation u=(m^2+n^2)/(1+mn) where (m,n)∈N.