Further results on the Diophantine equation x² + 16 ∙ 7b = yn when n is even.

This work extends the results for the Diophantine equation x² + 16 ∙ 7b = yn for n = 2r, where x,y, b,r ∈ Z+ . Earlier results classified the generators of solutions, which are the pair of integers (x,yr) into several categories and presented the general formula that determines the values of x and yr 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. [ABSTRACT FROM AUTHOR]