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

Author

Listed:
  • Saeree Wananiyakul
  • Prathomjit Khachorncharoenkul
  • Sor.Narth Hiransri
  • Palatip Khong-In
  • Kittipong Laipaporn

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.

Suggested Citation

  • Saeree Wananiyakul & Prathomjit Khachorncharoenkul & Sor.Narth Hiransri & Palatip Khong-In & Kittipong Laipaporn, 2026. "Finiteness and Explicit Solutions of 2x+pnqy=z2 for Certain Odd Primes," Journal of Mathematics, Hindawi, vol. 2026, pages 1-14, September.
    • Handle: RePEc:hin:jjmath:8840164
    DOI: 10.1155/jom/8840164
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