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I had a question in my mind from many years. we generally present the trivial solutions to Diophantine equations. Diophantine equations often have some sort of trivial solution (if you allow zero even Fermat’s equation has trivial solutions: $0^n+x^n=x^n$). I suppose the hard question is whether there are non-trivial solutions? The question of non-trivial solutions is already tough for polynomial Diophantine equations and requires detailed knowledge in algebraic and arithmetic geometry, modular forms, etc (see Wiles’ proof of Fermat’s last theorem).

What I am looking for, can we find non-trivial solutions to any polynomial Diophantine equations? If yes, what are the method so far existing?

Thanks in advance to all members of mse.

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    $\begingroup$ There was a fairly similar (although more focused) question on this site before: math.stackexchange.com/questions/13166/… $\endgroup$
    – Alex B.
    Apr 3, 2012 at 15:26
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    $\begingroup$ @gandhi: The solution to Hilbert's $10$th problem still leaves many interesting questions. Is there an algorithm for two-variable equations? Probably. For three-variable equations? For cubics in many variables? $\endgroup$ Apr 3, 2012 at 15:27
  • $\begingroup$ Usually, diophantine equations with variables in the exponent are even more difficult to solve as polynomial diophantine equations. I changes "usually always" to "often" since there are many diophantine equations without trivial solutions being interesting for number theoreticists , for example the Pillai conjecture. $\endgroup$
    – Peter
    Apr 30 at 11:20
  • $\begingroup$ Do you mean "polynomial Diophantine equations"? If so, then the post by MH is not an answer. $\endgroup$
    – user21820
    Apr 30 at 11:21
  • $\begingroup$ For a lot of equations, you can write parametrization. But usually getting solutions is an extremely difficult task. And when someone gets a formula, it is ignored. Because it contradicts the 10th hypothesis. You are not the first person to ask such a question - and you can say in advance that you will not like the answer anyway. People want a simple answer, but the reality is that the solution is very complicated. Not to mention solutions of systems of diophantine equations. $\endgroup$
    – individ
    Apr 30 at 11:22

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This question is Hilbert's 10th problem, from Hilbert's famous list of 23 problems promulgated 112 years ago in 1900.

The work of Julia Robinson, Martin Davis, Hillary Putnam, and Yuri Matijasevich, culminating in 1970, showed that it lacks an algorithmic solution.

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  • $\begingroup$ Sorry to say this, I think that there is some difference between diophantine equations and polynomial diophantine equations which OP is referring to . $\endgroup$
    – IDOK
    Apr 3, 2012 at 12:55
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    $\begingroup$ @Iyengar : You'll need to explain. What Hilbert's 10th problem deals with is polynomials $p(x_1,\ldots,x_n)$ in several variables with integer coefficients. The problem is to find integer solutions. These are "polynomial Diophantine equations" since $p$ is a polynomial. Your answer seems to assume that polynomial solutions are sought. But I see nothing int he posting clearly indicating that. $\endgroup$ Apr 3, 2012 at 15:16
  • $\begingroup$ See the comments above . $\endgroup$
    – IDOK
    Apr 4, 2012 at 11:26

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