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can you generalize the method of solving the equation of the type $2^x$ + $19^y$ = $z^2$ has a solution (3, 0, 3). Also, $8^x$ + $19^y$ = $z^2$ has no solution. But, $8^x$ + $17^y$ = $z^2$ has a solution (2, 1, 9). I think these there are very close to each other in means of prime base of each y term.

What is the important role by Catalan conjecture in this questions. Instead of trail and error can we write some general solutions of those equations. Can you discuss...plz?

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The @ does not work like this; you may want to log into whatever account you used for your previous question, and ask this in a comment to the answer you refer to. You should not begin a new question to engage in discussion with another user regarding their answer to a previous question. – Arturo Magidin Mar 21 '12 at 4:23
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Your title has two typos in two words: that's usually a bad idea... Please, do not address questions to specific users: if you spend five minutes browsing the site, you will notice that that is not how this site operates. (We do not write plZ here, by the way!) – Mariano Suárez-Alvarez♦ Mar 21 '12 at 4:24
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For those who wonder... – t.b. Mar 21 '12 at 4:24
Thanks t.b. I did wonder. – mixedmath♦ Mar 21 '12 at 4:25
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@Adam: Done. Anyone can still see the original post by visiting the revision history. I'll delete my previous comment, and this one after you see it (maybe "upvote" this comment so I know?) – The Chaz 2.0 Mar 22 '12 at 15:02
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closed as too localized by Arturo Magidin, Alex B., Asaf Karagila, t.b., The Chaz 2.0 Mar 22 '12 at 15:00

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1 Answer

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  1. The "Catalan conjecture" is no longer a conjecture, but a Theorem.

  2. I don't think the Catalan equation $x^m-y^n=1$ has much to do with the equations you ask about. What makes you think it has an "important role"?

  3. Search for "exponential diophantine equation" to get some idea of what techniques are known for working on the kind of equation you are asking about. There are almost as many techniques as equations, so m.se is probably not the place for an exhaustive answer to your question.

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