What makes an unsolved problem "interesting"?

Question

Beyond well-known unsolved problems like the Collatz conjecture or Recaman's sequence, one can trivially come up with problems in a similar vein (or perhaps an unusual infinite sum, or the irrationality of a number) that are also "unsolved" in the same sense. How do mathematicians decide which unsolved problems are worthwhile to pursue, i.e. if their solutions would have implications for other branches of mathematics or have other useful implications? Clearly not every unsolved problem, or a problem that's unsolvable with currently known mathematics, is of equal merit.

I'd assume that unsolved problems that are generalizable (that open the way to solving other unsolved problems) are more useful, but then again, isn't that also difficult to determine? In any case, there are far more challenging unsolved problems than mathematicians able to solve them.

Answer 1

Perhaps the greatest scholar of mathematical question posing, David Hilbert, had a number of guidelines for what made a good problem:

• They are not too easy and not too hard ("lest they 'mock' our attempts to solve them")
• They unify or connect apparently disparate branches of mathematics
• They lead to new problems
• They require the development of methods and techniques that have broad applicability

The relative weightings and judgements of these criteria can of course be subjective and differ among experts, but overall, among experts there is far more broad agreement than disagreement. Sometimes we don't realize a problem is important until after it is solved.

I consider it like fine art. There is no universally agreed upon definition of "beauty" or "importance" of a work of art (and that is a GOOD thing)... nevertheless among the educated there is remarkable general agreement.

Diego Velazquez's "Las meninas" is more important and interesting than my pencil self portrait.

The Riemann hypothesis is more important and interesting than any set of linear equations.

Incidentally, as you may know, in 1900 Hilbert posed his famous 23 outstanding problems in mathematics. Several of them are still unsolved and yet of great interest to mathematicians. Imagine YOU had to pose 20 problems of which many would not be solved in 121 years yet still be of interest to mathematicians.

I consider his posing of questions the greatest single instance of asking GOOD questions in the history of mathematics.

https://www.youtube.com/watch?v=PkcHstP6Ht0