I hereby propose as one of innumerable possible answers to this question: Hilbert's 10th problem.

<b>Doubtless it's an interesting problem</b>, to those who are interested in that sort of thing; otherwise Hilbert would not have included it in his list.  <b>But</b> to me, and again I suspect, to many, the answer is a lot more interesting than the question, partly, but not only, because it is surprising.

The problem is this: Is there an algorithm that given any polynomial in any finite number of variables with coefficients in $\mathbb Z$, correctly answers the question: is at least one tuple of integers a zero of this polynomial?

To understand the answer, let's establish some defintions:

 * A set $S$ of members of $\mathbb Z^n$ is <b>diophantine</b> if there is some $m\in\mathbb Z^+$ and some polynomial function $f$ in $m+n$ variables $y_1,\ldots,y_m,x_1,\ldots,x_n$ with coefficients in $\mathbb Z$ such that $(x_1,\ldots,x_n)\in S$ if and only if $\exists y_1,\ldots,y_m\in\mathbb Z\  f(y_1,\ldots,y_m,x_1,\ldots,x_n)=0$.
 * A set $S$ of members of $\mathbb Z^n$ is <b>decidable</b> if there is some algorithm that, given a member of $\mathbb Z^n$ correctly answers the question: Is this a member of $S$?
 * A set $S$ of members of $\mathbb Z^n$ is <b>semi-decidable</b> if there is some algorithm that, given a member of $\mathbb Z^n$, runs forever if the input is not a member of $S$, and ultimately halts if it is a member of $S$.

Obviously a set is decidable if and only if both the set and its complement are semi-decidable.  The existence of semidecidable sets that are not decidable was discovered in the 1930s by several people working independently (I think including Stephen Kleene, Alan Turing, Alonzo Church and maybe others?) and some of them are noteworthy sets, e.g. the set of all satisfiable formulas in first-order logic.

Obviously all diophantine sets are semi-decidable.

The result that laid Hilbert's 10th problem to rest is <b>Matiyasevich's theorem:</b>

<b>All semi-decidable sets are diophantine.</b>

An immediate corollary is that no algorithm of the kind sought by Hilbert can exist.

In 1970, Yuri Matiyasevich finished off the proof, which had been worked on over a couple of decades by Julia Robinson, Martin Davis, and Hillary Putnam.