Further results on the Diophantine equation x^2+16·7^b = y^n when n is even - Yow, Kai Siong; Sapar, Siti Hasana; Pham, Hoa (Prince of Songkla University, 2024)
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 in which integral solutions exist.