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On the Diophantine Equation a^x+(a-1)^y=z^2.
Apisit Pakapongpun, Rakporn Dokchan, Natdanai Chailangka
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Source: Crossref
Published: Jan 1, 2026
DOI: 10.69793/ijmcs/04.2026/pdc
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Using modulo properties, we prove that the exponential Diophantine equation a^x+(a-1)^y=z^2 has no solution in non-negative integers x, y, z when a=24N+5, where both N and a are prime. This extends the results of the specific cases 29^x+28^y=z^2 and 3^x+52^y=z^2$ to an infinite family of primes of the form 24N+5.
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