From the results of the Hilbert's tenth problem, does it follows that there exist infinitely numbers of Diophantine equations where it is impossible to determine whether or not there are integer solutions?

Conversly, is it true that there exist infinite number of Diophantine equations where it can be determine rather or not there are integer solutions?

If both are true, why do people still have the reason to study it?

Hilberts 10th Problem (wiki) is:

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.


1 Answer 1


There are algorithms for determining the solvability in integers of certain classes of Diophantine equations. For example, we can deal quite easily with linear equations in arbitrarily many variables, and there are infinitely many of those. There is also an algorithm for quadratic Diophantine equations in arbitrarily many variables, though this is less obvious. The answer for cubics is not known. There is no general algorithm for determining the solvability of quartic equations in arbitrarily many variables.

There is a polynomial $P(w,x_1,x_2,x_n)$ with integer coefficients such that there is no algorithm that will determine, given input $w$, whether there are integers $x_1,x_2,\dots,x_n$ such that $P(w,x_1,x_2,\dots,x_n)=0$. As a consequence of this, there are infinitely many $w$ such that the equation has no solution, but the fact that it has no solution is not provable in first-order Peano arithmetic. A similar remark could be made about ZFC. I do not think that this result can be equated with "impossible to determine," but it goes some distance toward that.

The above results are very beautiful. But they do not necessarily have a direct connection with classical questions in Diophantine equations, such as questions abut elliptic curves. The work on Hilbert's $10$th problem in fact produced interesting number-theoretic results, and generated new number-theoretic problems. It has enriched Number Theory, and in no way diminished it.


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