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Hello all!

In the pursuit of a minor research problem I was pointed in the direction of an interesting result in the realm of Diophantine Analysis. The content of the result follows:

$\frac{ax^{n+2l}-1}{ax^{n}-1} = y^2$ has a solution in the natural numbers for $a$, $x$, $n$, and $l$, with $x>1$ and $y$ a rational number, if and only if the following conditions are met:

• $2|l$
• $a = \frac{3^{l-1}+1}{4}$
• $x=3$
• $n=1$
• $y=\pm (3^l +2)$

I am looking to make a slight alteration to this theorem for use in my research (though the term should be interpreted lightly), by changing $y^2$ to $3y^2$. However, after conferring with one of my professors, I was told such a theorem may already exist!

Does anyone know of this (or these) results, and would be willing to suggest a reference? If no such theorem exists, does anyone have any pointers on a good approach to this problem?

-Richard Voepel

[EDIT]

I first read this result from one of my professor's papers, though I was told directly this was not the first paper to provide such a result. Here is a link.

http://www.math.sc.edu/~filaseta/papers/DiophantinePaper2006.pdf

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# Slight Alteration to a Diophantine Result

Hello all!

In the pursuit of a minor research problem I was pointed in the direction of an interesting result in the realm of Diophantine Analysis. The content of the result follows:

$\frac{ax^{n+2l}-1}{ax^{n}-1} = y^2$ has a solution in the natural numbers for $a$, $x$, $n$, and $l$, with $x>1$ and $y$ a rational number, if and only if the following conditions are met:

• $2|l$
• $a = \frac{3^{l-1}+1}{4}$
• $x=3$
• $n=1$
• $y=\pm (3^l +2)$

I am looking to make a slight alteration to this theorem for use in my research (though the term should be interpreted lightly), by changing $y^2$ to $3y^2$. However, after conferring with one of my professors, I was told such a theorem may already exist!

Does anyone know of this (or these) results, and would be willing to suggest a reference? If no such theorem exists, does anyone have any pointers on a good approach to this problem?