Finiteness and Explicit Solutions of 2x+pnqy=z2 for Certain Odd Primes

Saeree Wananiyakul, Prathomjit Khachorncharoenkul, Sor.Narth Hiransri, Palatip Khong-In and Kittipong Laipaporn

Journal of Mathematics, 2026, vol. 2026, 1-14

Abstract: We investigate all nonnegative integer solutions x,y,z of the exponential Diophantine equation 2x+pnqy=z2, where p and q are fixed odd primes, and n is a fixed positive integer. Using 2-adic valuation techniques and known results on exponential equations, we classify all solutions explicitly under some conditions on p,q, and n. In particular, we show that under this condition, the equation admits at most two solutions and has exactly two solutions if and only if pn,q=3,5,7,5, or 9,17. In addition, we also show that there are infinitely many pn,q such that the equation has exactly one solution and also there are infinitely many pn,q such that the equation has no solution.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:8840164

DOI: 10.1155/jom/8840164

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