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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.

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3 Answers 3

up vote 4 down vote accepted

In a recent manuscript Apoloniusz Tyszka showed that for $n\ge 12$, subsets $S$ of the restricted equations $E_n$ exist which have infinitely many integer solutions, but for which the smallest component of every integer solution of $S$ is at least $2^{2^{n-1}}+1$ in absolute value. $E_n$ is defined as the set of all equations of the form $x_i=1$ for $i=1,2,\dots,n$, as well as $x_i=x_j+x_k$ and $x_i=x_j.x_k$ for $i,j,k=1,2,\dots,n$. Any polynomial equation can be expressed using some subset $S$ of $E_n$, for some large enough $n$ that depends on the form of the equation.

Not only the variables in the original equation exceed the $2^{2^{n-1}}$ bound, but so do all the intermediate variables that appear in $S$.

  • Apoloniusz Tyszka, Small systems of Diophantine equations which have only very large integer solutions, 2011. arXiv:1102.4122
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Modestly huge numbers can come from simple equations, such as Pell Equations. Already the smallest non-trivial solution of $x^2-61y^2=1$ is quite large, with $|x|$ and $|y|$ around $10^9$.

The following result, connected with the solution of Hilbert's 10th Problem, shows that that the rate of growth of solutions can be truly enormous. There is a polynomial $P(e,x,y_1,y_2,\dots,y_k)$ with integer coefficients such that for any computable (or even recursively enumerable) set $S$ of positive integers, there is an $e=e(S)$ such that $x \in S$ if and only if the Diophantine Equation $P(e,x,y_1,y_2,\dots,y_k)=0$ has a solution.

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To be more explicit, the result user6312 cites shows that there does not exist a computable bound, since a computable bound would allow us to solve arbitrary Diophantine equations (which we can't). –  Qiaochu Yuan Apr 16 '11 at 16:57
    
Yes certainly this holds for large enough degree and numbers of variables. Maybe there are some computable results for smaller degrees? –  quanta Apr 16 '11 at 18:20

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.

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