From the past, we had Fermat's Last Theorem. Looking for ones that are not solved yet? How about "Collatz conjecture
"? I think there are more complex ones, but yeah, not so simple to explain.
from everything I've read, collatz has not merely resisted proof, but anything resembling progress. Erdos said that mathematics is not yet ready for such problems.
Conway showed that the generalized Collatz conjecture (recurrences with arbitrary cases dependent on the modulus) is computationally undecidable (halting problem reduces to it). The choice of modulus doesn't even need to be that big to get this result, only ~6500 or so. As far as I know, this is the only substantial result in either direction for this problem.
I suspect it is due to the lack of application for the solution.
I analyzed the problem for a while and theorized ways to collaboratively contribute to the solution. I settled on submitting an integer sequence to OEIS.
True. And there could be many more ridiculously simple problems with hard or no applications (at least for now) we may not be able to solve or progress at all. Math as we know and practice might still fall short. I, for one, be happy that simple to state but solved, Godel's Incompleteness theorem, about the imperfection of the tools, had set the things firmly on stone.
Solving them or progress? May be I don't care, but these little problem, simple ones does encourage one to start thinking about solving problems, they make math accessible to masses.