TITLE: Naming big numbers with Diophantine equations
QUESTION [6 upvotes]: I haven't noticed any obvious bound on the size of the solutions of a Diophantine equation.
There may be some results for quadratics such as $ax^2+by^2+cz^2=0$ has a solution bounded by $abc$ (that is probably wrong but I read something along that lines?).

What results are there for small degrees, small numbers of variables?
What examples of there of huge numbers defined by small Diophantine equations?


edit: Related Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution? but I haven't read it yet.

REPLY [3 votes]: An integer is said to be $\it congruent$ if it is the area of a right-triangle with rational sides. This can be rephrased as two quadratic diophantine equations in 4 variables. There are smallish congruent numbers (less than 200) for which the smallest solutons involve 100 digits or thereabouts. 
More examples may be found in a discussion of Eventual Counterexamples at MathOverflow.