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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Mohaghegh Ardabili
2023-12-01
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| Series: | Journal of Hyperstructures |
| Subjects: | |
| Online Access: | https://jhs.uma.ac.ir/article_2591_aa4628780030854e2945e50ed8c2857a.pdf |
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| Summary: | 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. |
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| ISSN: | 2251-8436 2322-1666 |