Slight Alteration to a Diophantine Result - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:35:05Z http://mathoverflow.net/feeds/question/61775 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61775/slight-alteration-to-a-diophantine-result Slight Alteration to a Diophantine Result Richard Voepel 2011-04-15T01:56:04Z 2011-04-17T06:15:28Z <p>Hello all!</p> <p>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:</p> <p>$\frac{ax^{n+2l}-1}{ax^{n}-1} = y^2$ 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:</p> <ul> <li>$2|l$</li> <li>$a = \frac{3^{l-1}+1}{4}$</li> <li>$x=3$</li> <li>$n=1$</li> <li>$y=\pm (3^l +2)$</li> </ul> <p>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!</p> <p>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?</p> <p>Thank you all in advance for your time and help!</p> <p>-Richard Voepel</p> <p>[EDIT]</p> <p>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.</p> <p><a href="http://www.math.sc.edu/~filaseta/papers/DiophantinePaper2006.pdf" rel="nofollow">http://www.math.sc.edu/~filaseta/papers/DiophantinePaper2006.pdf</a></p>