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It seems like there's been an explosion of (exponential) diophantine equations with straightforward solutions lately and it would be great to have an example at hand of how such simple equations can have dramatically more complex solutions. I'm familiar with the classical example of Pell's equation $x^2-61y^2=1$ where the minimal solution is $x=1766319049, y=226153980$, but it seems like the expressive power of exponentiation ought to enable even more dramatic examples (e.g. something like an equation in 3 or 4 terms of single-digit height with minimal solutions in the dozens of digits). Does anyone know of simple examples comparable to the above?

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This recent article discusses solutions to $x^3+y^3=n$ in rationals with some smallest solutions in the dozens of digits for small n. It ends with the example that $x^3+y^3=4981z^3$ in positive integers requires >16million digits for the smallest solution.

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Nice, but I interpret "exponential diophantine equation" to mean at least one of the variables is in the exponent. –  Gerry Myerson Jun 5 '12 at 1:43

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