Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ

Solutions to the diophantine equation x² + 16∙7ᵇ = y²ʳ

We present a method of determining integral solutions to the equation x2 + 16 ∙ 7b = y2r, where x, y, b, r ∈ ℤ+. We observe that the results can be classified into several categories. Under each category, a general formula is obtained using the geometric progression method. We then provide the bound...

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Bibliographic Details
Main Authors: Yow, Kai Siong, Sapar, Siti Hasana, Low, Cheng Yaw
Other Authors: School of Computer Science and Engineering
Format: Article
Language:English
Published: 2023
Subjects:
Engineering::Computer science and engineering
Diophantine Equation
Integral Solution
Online Access:https://hdl.handle.net/10356/164523
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Institution: Nanyang Technological University
Language: English
Description
Summary:We present a method of determining integral solutions to the equation x2 + 16 ∙ 7b = y2r, where x, y, b, r ∈ ℤ+. We observe that the results can be classified into several categories. Under each category, a general formula is obtained using the geometric progression method. 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 the respective values of x and yr. We also highlight two special cases where different formulae are needed to represent their integral solutions.

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