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In solving equations, we first look into whether solutions exist or not. If they do, we analyze if the solutions are finite or infinite in number. If infinite, we give solutions in general form. With Diophantine equations, we search for solutions in positive integers.

Now my question is how to proceed as above for the following problems:

  1. $\displaystyle \frac{x^y - y^x}{x - y} = z^2$; I got one solution $(x, y, z) = (2, 3, 1)$.

  2. $x^3 + y^2 - x^2y + x^2y^3 - y^2x = x + y$; I got one solution $(x, y) = (1, 1)$.

The above solutions I got by trial-and-error. I would like to know the following:

  1. How to know if solutions exist or not for above equations?

  2. If they do, how to know if there are finitely or infinitely many?

  3. If finite how to find all solutions?

  4. If infinite how to characterize all solutions by a general formula?

  5. Is there any theory or particular method generalizing both the problems?

Thank you,

  • 1
    Non-linear (let alone exponential, like your first equation) diophantine equations are notoriously difficult: just look at the problem of determining that the Fermat equation has no nontrivial solutions for specific exponents, or Catalan's conjecture.2012-03-12
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    I am also interested to know the finding solutions of the first question. Whereas, the solution of second question is wrong. By inspection the solution given for first question is correct. But, how to get this solution without inspection by trail and error method? Is there any member, who had an idea for finding solutions for first problem.2012-03-13

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