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On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m | ||
| Journal of Hyperstructures | ||
| دوره 11، شماره 2، اسفند 2022، صفحه 329-337 اصل مقاله (273.59 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22098/jhs.2023.2591 | ||
| نویسندگان | ||
| Mridul Dutta* 1؛ Padma Bhushan Borah2 | ||
| 1Department of Mathematics, Dudhnoi College, P.O. Dudhnoi, Goalpara, Assam, India | ||
| 2Department of Mathematics, Gauhati University, Guwahat, Assam, India | ||
| چکیده | ||
| It is well known that the exponential Diophantine equation 2x+ 1=z2 has the unique solution x=3 and z=3 in non-negative integers, which is closely related to the Catlan's conjecture. In this paper, we show that for m∈N, m>1, the exponential Diophantine equation 2x+m2y=z2 admits a solution in positive integers (x, y,z) if and only if m=2αMn, α≠0 for some Mersenne number Mn. When m=2αMn, α≠0, the unique solution is (x,y,z)=(2+n+2α,1, 2α(2n+1)). Finally, we conclude with certain examples and non-examples alike! The novelty of the paper is that we mainly use elementary methods to solve a particular class of exponential Diophantine equations. | ||
| کلیدواژه‌ها | ||
| Mersenne numbers؛ Catalan's Conjecture؛ Exponential Diophantine equations | ||
| مراجع | ||
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