| dc.contributor.author | Kittipong Laipaporn | |
| dc.contributor.author | Saeree Wananiyakul | |
| dc.contributor.author | Prathomjit Khachorncharoenkul | |
| dc.contributor.other | Center of Excellence for Ecoinformatics, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand | |
| dc.contributor.other | Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand | |
| dc.contributor.other | Center of Excellence for Ecoinformatics, School of Science, Walailak University, Nakhon Si Thammarat 80160, Thailand | |
| dc.date.accessioned | 2025-08-27T02:32:32Z | |
| dc.date.accessioned | 2025-10-08T08:04:27Z | |
| dc.date.available | 2025-10-08T08:04:27Z | |
| dc.date.issued | 2025-07 | |
| dc.identifier.uri | https://www.aimspress.com/article/doi/10.3934/math.2025704 | |
| dc.identifier.uri | http://digilib.fisipol.ugm.ac.id/repo/handle/15717717/35583 | |
| dc.description.abstract | Over the past decade, significant research has been conducted on the equation $ a^x+b^y = z^2 $ under various conditions imposed on $ a $ and $ b $ or on $ x $ and $ y $. Most studies focus on conditions where the equation has no solution, while some explore cases with infinitely many solutions, often considering scenarios where $ x $ or $ y $ is even. Motivated by this line of inquiry, we have been inspired to investigate and analyze equations of the form $ p^x+ q^{2y} = z^{2 n} $ for two distinct primes $ p $ and $ q $, and to present explicit forms of their solutions $ (p, x, q, y, z, n) $. Recent studies on the exponential Diophantine equation $ p^x+q^y = z^2 $, where $ p $ and $ q $ are primes, have addressed cases where $ p = 2 $ or $ p\equiv q\pmod 4 $. In this paper, we address the case where $ p\not\equiv q\pmod 4 $ and $ y $ is even. In addition, we explore special cases where $ z $ is the prime and provide the complete set of solutions for $ p^x+q^{2y} = z^{2n} $. We also show that the equation has no solution when $ \{2, 3\}\nsubseteq\{p, q, z\} $. In other words, we provide almost explicit solutions to $ p^x+ q^{y} = z^{2 n} $ except for the case where both $ x $ and $ y $ are odd. | |
| dc.language.iso | EN | |
| dc.publisher | AIMS Press | |
| dc.subject.lcc | Mathematics | |
| dc.title | Explicit solutions and non-solutions for the Diophantine equation $ p^x+ q^{2y} = z^{2 n} $ involving primes $ p\not\equiv q \pmod 4 $ | |
| dc.type | Article | |
| dc.description.keywords | diophantine equation | |
| dc.description.keywords | catalan's conjecture | |
| dc.description.keywords | legendre symbol | |
| dc.description.pages | 15720-15736 | |
| dc.description.doi | 10.3934/math.2025704 | |
| dc.title.journal | AIMS Mathematics | |
| dc.identifier.e-issn | 2473-6988 | |
| dc.identifier.oai | deb6c900f36d430ea90473f82737fc5a | |
| dc.journal.info | Volume 10, Issue 7 | |
