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I am looking for results on the equation $$ax^n-by^m=1 \tag 1 $$ where $\gcd(m,n)=1$ and $a,b,n,m$ are constants.

I found literature for $ax^n-by^n=1$ (R. A. Mollin, D. T. Walker) but couldn't find anything on equation $(1)$.

The closest I got is Pillai's conjecture which is a generalization of Catalan's conjecture.

Does $ax^n-by^m=1$ has infinite solution? If it does how do we find them?

Please provide related literature/reference if possible.

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A lot is known about such diophantine equations of the form $F(x,y)=0$ for a polynomial $F$, including results about when the number of solutons can be infinite (this happens in very rare cases that don't include the ones you are interested in) as well as efficient methods to find the solutions. I would recommend looking at the paper "The Diophantine Equation $f(x)=g(y)$" by Y. F. Bilu and R. F. Tichy. It has an extensive list of references that cover most classical results on such equations.

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  • $\begingroup$ Thannks but could you specifically write on the equation I mentioned above plz? what we know about it? The paper deals with the general case, so it is hard to extract, it would be helpful if you break it down for the above mentioned equation and provide known reaults on it. $\endgroup$
    – Consider Non-Trivial Cases
    Commented Dec 31, 2019 at 16:46
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    $\begingroup$ @Andrew It says that if your constants satisfy $a,b\neq 0, \gcd(m,n)=1, m,n\geq 2$, then the equation has finitely many solutions. $\endgroup$
    – Gjergji Zaimi
    Commented Dec 31, 2019 at 16:52
  • $\begingroup$ if possible plz provide the page numbers related to the equation and ur last comment. $\endgroup$
    – Consider Non-Trivial Cases
    Commented Dec 31, 2019 at 16:55
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    $\begingroup$ @Andrew did you read the paper? If you're having trouble with relating the contents to your specific interest, then I would recommend asking questions on math.stackexchange $\endgroup$
    – Gjergji Zaimi
    Commented Dec 31, 2019 at 17:07
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    $\begingroup$ @Andrew, the paper mentioned has been offered as a starting point to the literature, with a list of references for you to follow. In addition, the main Theorem says when your equation has infinitely many solutions (this is primarily when at least one of the exponents are small). You will have to do some more work yourself to get the information. For the expected audience of this forum, Gjergji has answered your question. Gerhard "We're All About Mathematical Research" Paseman, 2019.12.31. $\endgroup$
    – Gerhard Paseman
    Commented Dec 31, 2019 at 19:06

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