On the Diophantine Equations p^x+(p+30)^y=z^2
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Abstract
This research investigates the non-negative integer solutions to the Diophantine equation , specifically for the case where
is not a multiple of
. Under this condition, we prove that the equation has a unique non-negative integer solution at
. The proof is based on the analysis of parity, divisibility properties, modular arithmetic, and Mihăilescu's Theorem. The result indicates that imposing specific conditions on variables can lead to a definite solution for a complex Diophantine equation, suggesting a potential approach for studying the equation in more general forms.
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