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

## Authors

• Kai Siong Yow ᵃ Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia; ᵇ School of Computer Science and Engineering, College of Engineering, Nanyang Technological University, Singapore
• Siti Hasana Sapar Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia
• Cheng Yaw Low Data Science Group, Institute for Basic Science, Expo-ro 55, Yuseong-gu, Daejeon, 34126, South Korea

## Keywords:

Diophantine equation, integral solution, geometric progression, polynomial

## 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.

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