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William S. Gayo, Jr.,
On the Exponential Diophantine Equation 29^x+28^y=z^2.
Int. J. Math. Comput. Sci., 20, no. 2, (2025), 517-519

DOI:

https://doi.org/10.69793/ijmcs/02.2025/william

Keywords and phrases:

Diophantine equation, integer solutions, modular arithmetic, primes.

Abstract:

We first show the condition for which the Diophantine equation w^x+(w-1)^y=z^2, where w is a positive integer of the form 24N+5, may admit nonnegative integer solutions. Then, using this condition and the modular arithmetic method, we demonstrate that the Diophantine equation 29^x+28^y=z^2 has no solution in nonnegative integers.