Solution of the exponential Diophantine Equation
Keywords:
Diophantine equations, integer solutions, number theoryAbstract
This study investigates the exponential Diophantine equation where p is a prime number and n, x, y, z are non-negative integers, subject to the modular condition
The primary objective is to determine all non-negative integer solutions of this equation by employing quadratic residue theory, modular arithmetic, and its connections to Pell-type equations.
The results demonstrate that the equation admits a unique non-negative integer solution given by
For all other values of p, no non-negative integer solutions exist, and it can be rigorously proven that cannot be a perfect square outside this solution. These findings provide a clear classification of the solution set structure and offer theoretical insights beneficial for further studies on exponential Diophantine equations, including potential applications in computational number theory and cryptographic systems.
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