Abstract

This paper investigates the Diophantine equation p r + (p + 1) s = z 2 Where p > 3, s ≥ 3 , z is an even integer. The focus of the study is to establish rigorous results concerning the existence of solutions within this specific parameter space. The main result presented in this paper demonstrates the absence of solutions under the stated conditions. The proof employs mathematical techniques to systematically address the case when the prime p exceeds 3, and the exponent s is equal to or greater than 2, while requiring the solution to conform to the constraint of an even z. This work contributes to the understanding of the solvability of the given Diophantine equation and provides valuable insights into the interplay between prime powers and the resulting solutions.