Further results on the Diophantine equation 𝑥 2 + 16 ∙ 7 𝑏 = 𝑦 𝑛 when 𝑛 is even

Abstract

This work extends the results for the Diophantine equation 𝑥 2 + 16 ∙ 7 𝑏 = 𝑦 𝑛 for 𝑛 = 2𝑟, where 𝑥, 𝑦, 𝑏, 𝑟 ∈ ℤ + . Earlier results classified the generators of solutions, which are the pair of integers (𝑥, 𝑦 𝑟 ), into several categories and presented the general formula that determines the values of 𝑥 and 𝑦 𝑟 for the respective category. The lower bound for the number of non-negative integral solutions associated with each 𝑏 is also provided. We now extend the results and prove the necessary and sufficient conditions required to obtain integral solutions 𝑥 and 𝑦 to the equation, by considering various scenarios based on the parity of 𝑏. We also determine the values of 𝑛 for which integral solutions exist.

Keywords

• diophantine equation
• polynomial
• generator
• integral solution