Solutions to the Diophantine Equation x²+16⋅7ᵇ=y²ʳ

Abstract

We present a method of determining integral solutions to the equation x^2 + 16.7^b = y^2r, where x,y,b,r \in Z^+. We observe that the results can be classified into several categories. Under each category, a general formula is obtained by using the method of geometric progression. We then provide the bound for the number of non-negative integral solutions associated with each b. Lastly, the general formula for each of the categories is obtained and presented to determine respective values of x and y^r. We also highlight two special cases where different formulae are needed to represent their integral solutions.