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I posted this question on Math.SE and it has been a while that no non-trivial hint has been suggested. I hope MOers will have something to say or will solve it entirely. Here is the question copied verbatim:

I stumbled upon this number theory problem while I was solving another problem. Here is the equation: $$3^kn + 3^{k-1} + 2^m(3^{k-1} + 2h) = 2^{m+l}n$$ where $k \geq 3, h,l,m,n\in\mathbb{N}$, $n$ is oddand , not a multiple of $3$. 3$ and $n\geq 7$. My impression is that it does not have a solution. However, I have not progressed on the problem anymore than that. Could you please help?

Thanks.
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I posted this question on Math.SE and it has been a while that no non-trivial hint has been suggested. I hope MOers will have something to say or will solve it entirely. Here is the question copied verbatim:

I stumbled upon this number theory problem while I was solving another problem. Here is the equation: $$3^kn + 3^{k-1} + 2^m(3^{k-1} + 2h) = 2^{m+l}n$$ where $k \geq 3, h,l,m,n\in\mathbb{N}$, $n$ is odd and not a multiple of $3$. My impression is that it does not have a solution. However, I have not progressed on the problem anymore than that. Could you please help? Do you smell anything in this equation?

Thanks.
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Solving in positive integers an equation containing exponentials

I posted this question on Math.SE and it has been a while that no non-trivial hint has been suggested. I hope MOers will have something to say or will solve it entirely. Here is the question copied verbatim:

I stumbled upon this number theory problem while I was solving another problem. Here is the equation: $$3^kn + 3^{k-1} + 2^m(3^{k-1} + 2h) = 2^{m+l}n$$ where $k \geq 3, h,l,m,n\in\mathbb{N}$, $n$ is odd and not a multiple of $3$. My impression is that it does not have a solution. However, I have not progressed on the problem anymore than that. Could you please help? Do you smell anything in this equation?

Thanks.