On the solution of the exponential Diophantine equation 2x+m2y=z2, for any positive integer m

Document Type : Research Paper

Authors

1 Department of Mathematics, Dudhnoi College, P.O. Dudhnoi, Goalpara, Assam, India

2 Department of Mathematics, Gauhati University, Guwahat, Assam, India

Abstract

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.
 

Keywords

References

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Journal of Hyperstructures
Volume 11, Issue 2
December 2022
Pages 329-337

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