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Landau's four problems are now over a century old (1912), and each still unsolved. This seems remarkable, even though he was not the originating author all four (maybe only the 4th?). Still, he isolated and listed them as challenges.

Of course Hilbert's $23$ problems (1900) have been hugely influential, but many have been resolved in some form; perhaps $4$ sharply defined problems remain completely unresolved. William Thurston's more focussed $24$ problems (1982) are largely resolved: Thurston's 24 questions: All settled?. Steve Smale's 18 problems (1998) are perhaps half solved. Geoffrey Shephard's 1968 list of $20$ questions was narrowly focused on convex polyhedra.

What other such lists have mathematicians publicized? Is there any single researcher's list comparable to Landau's in duration remaining open?

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    $\begingroup$ Landau was trying to list "unattackable" problems, whereas Hilbert and Thurston were trying to spur research by stating hard but hopefully solvable problems. So I don't find it so remarkable that Landau's list remains unsolved. If we put our minds to it, it should be pretty easy to list problems that are likely to remain unsolved for 100 years, or even forever. For example, in Richard Stanley's Enumerative Combinatorics there is an exercise asking if there are infinitely many $n$ for which $f(n)$ is a palindrome in base ten, where $f(n)$ is the number of nonisomorphic posets on $n$ elements. $\endgroup$
    – Timothy Chow
    Commented Dec 10, 2022 at 1:27
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    $\begingroup$ I guess part of this is about the mathematician collecting together a single list; perhaps not quite that: en.wikipedia.org/wiki/Kaplansky%27s_conjectures $\endgroup$
    – Will Jagy
    Commented Dec 10, 2022 at 1:27
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    $\begingroup$ @TimothyChow Well, they are not as "unattackable" as they seemed to Landau and they did spur some interesting research. Of course, we do not have a gap of 2 between the primes, just 246; we do not have primes between $n^2$ and $(n+1)^2$, just between $n^3$ and $(n+1)^3$, we do not have a sum of two primes for evens, just a sum of 3 primes for odds and all of those were quite exciting mathematical achievements. So, while it doesn't take a genius to set up an interesting problem (much less to advertise one), I wouldn't say that Landau's list was less influential than Hilbert's. $\endgroup$
    – fedja
    Commented Dec 10, 2022 at 2:23
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    $\begingroup$ @fedja Was Landau's list really influential? The problems were well known before Landau, weren't they? I must confess that I had never heard of Landau's list before reading this MO question, even though I've know about all four problems for ages. A lot of Hilbert's problems, by contrast, were either originated by him or not at all well known before. $\endgroup$
    – Timothy Chow
    Commented Dec 10, 2022 at 2:34
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    $\begingroup$ @JoshuaZ Yes, that's a fair point; however, certainly some of Hilbert's problems are commonly referred to as such: Hilbert's tenth problem, Hilbert's thirteenth problem, Hilbert's sixteenth problem, etc. In other cases, the paper solving the problem mentions Hilbert explicitly; e.g., Artin's paper solving Hilbert's seventeenth problem starts off immediately by citing Hilbert. Hilbert's second problem is not always referred to by number, but it is widely known that "Hilbert's program" was an effort to prove the consistency of mathematics. $\endgroup$
    – Timothy Chow
    Commented Dec 11, 2022 at 2:21

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