Non-negative Solutions of the Nonlinear Diophantine Equation (8^n)^x + p^y=z^2 for Some Prime Number p

In this paper, we explain all non-negative integer solutions for the nonlinear Diophantine equation of type 8x + py = z2 when p is an arbitrary odd prime number and incongruent with 1 modulo 8. Then, we apply the result to describe all non-negative integer solutions for the equation (8n)x + py = z2 when n ≥ 2. The results presented in this paper generalize and extend many results announced by other authors. HIGHLIGHTS Studying a new series of the equation 8x + py = z2 when p is prime which is not congruent to 1 modulo 8 Describing all non-negative integer solutions of the equation (8n)x + py = z2 which is a generalization of the equation 8x + py = z2 The equation 8x + py = z2 has at most 3 non-negative integer solutions when p is congruent to 1 modulo 8 and p ≠ 17