On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m
It is well known that the exponential Diophantine equation 2x+ 1=z2 has the unique solution x=3 and z=3 innon-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...
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University of Mohaghegh Ardabili
2023-12-01
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| Series: | Journal of Hyperstructures |
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| Online Access: | https://jhs.uma.ac.ir/article_2591_aa4628780030854e2945e50ed8c2857a.pdf |
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| author | Mridul Dutta Padma Bhushan Borah |
| author_facet | Mridul Dutta Padma Bhushan Borah |
| author_sort | Mridul Dutta |
| collection | DOAJ |
| description | It is well known that the exponential Diophantine equation 2x+ 1=z2 has the unique solution x=3 and z=3 innon-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. |
| format | Article |
| id | doaj-art-c5e8ae0eb0db4759a4ed879da7079db0 |
| institution | Matheson Library |
| issn | 2251-8436 2322-1666 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | University of Mohaghegh Ardabili |
| record_format | Article |
| series | Journal of Hyperstructures |
| spelling | doaj-art-c5e8ae0eb0db4759a4ed879da7079db02025-07-09T08:39:32ZengUniversity of Mohaghegh ArdabiliJournal of Hyperstructures2251-84362322-16662023-12-0111232933710.22098/jhs.2023.25912591On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer mMridul Dutta0Padma Bhushan Borah1Department of Mathematics, Dudhnoi College, P.O. Dudhnoi, Goalpara, Assam, IndiaDepartment of Mathematics, Gauhati University, Guwahat, Assam, IndiaIt is well known that the exponential Diophantine equation 2x+ 1=z2 has the unique solution x=3 and z=3 innon-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.https://jhs.uma.ac.ir/article_2591_aa4628780030854e2945e50ed8c2857a.pdfmersenne numberscatalan's conjectureexponential diophantine equations |
| spellingShingle | Mridul Dutta Padma Bhushan Borah On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m Journal of Hyperstructures mersenne numbers catalan's conjecture exponential diophantine equations |
| title | On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m |
| title_full | On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m |
| title_fullStr | On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m |
| title_full_unstemmed | On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m |
| title_short | On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m |
| title_sort | on the solution of the exponential diophantine equation 2x m2y z2 for any positive integer m |
| topic | mersenne numbers catalan's conjecture exponential diophantine equations |
| url | https://jhs.uma.ac.ir/article_2591_aa4628780030854e2945e50ed8c2857a.pdf |
| work_keys_str_mv | AT mriduldutta onthesolutionoftheexponentialdiophantineequation2xm2yz2foranypositiveintegerm AT padmabhushanborah onthesolutionoftheexponentialdiophantineequation2xm2yz2foranypositiveintegerm |