Timeline for Landau's century-old problems: Anything comparable?
Current License: CC BY-SA 4.0
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| Dec 11, 2022 at 19:16 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Dec 11, 2022 at 13:41 | comment | added | Timothy Chow | I posted a question on the History of Science and Mathematics StackExchange asking about how influential Landau's list was. | |
| Dec 11, 2022 at 8:58 | history | edited | Alex M. |
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| Dec 11, 2022 at 2:37 | comment | added | JoshuaZ | @TimothyChow That's a good point. And for that matter, when I asked a question here related to Hilbert's Tenth about a year ago, i just called it that without batting an eye. So it may be that RH is not referred to that way because it is so famous on its own. | |
| Dec 11, 2022 at 2:21 | comment | added | Timothy Chow | @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. | |
| Dec 11, 2022 at 2:04 | comment | added | JoshuaZ | @TimothyChow Also relevant, how often do papers related to the more famous Hilbert problems mention that the problems are on Hilbert's list. For example, it seems very rare that paper dealing with RH mentions it being one of Hilbert's problems. | |
| Dec 11, 2022 at 1:26 | history | edited | Joseph O'Rourke | CC BY-SA 4.0 |
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| Dec 10, 2022 at 14:29 | comment | added | Timothy Chow | I spent a few minutes trying to find evidence that Landau's list was influential. Since Goldbach and twin primes were also on Hilbert's list, it would be hard to isolate Landau's influence from Hilbert's, unless an author mentioned Landau by name, so I focused on the other two. I looked at some famous papers proving partial results, and they didn't mention Landau's list. Of course this doesn't prove much. Are there any number-theory textbooks that explicitly mention Landau's list? | |
| Dec 10, 2022 at 8:04 | comment | added | user44143 | @fedja, Hilbert’s contemporaries found his problems idiosyncratic: axiomatic and unapplied, a contrast to Poincaré’s tastes. See Grattan-Guinness’s sideways looks, eg quoting Hilbert’s student Blumenthal from 1935: “Quite few [of Hilbert’s problems] stem from the general situation of mathematics or from the problem-contexts of other researchers”. ams.org/notices/200007/fea-grattan.pdf | |
| Dec 10, 2022 at 2:40 | comment | added | fedja | @TimothyChow I mean the collection of the problems was influential, not Landau's putting them together. Were all Hilbert problems unknown before Hilbert? Besides, I was born well after 1912, so if I knew something for all my life, it doesn't mean it was equally popular and well-known before 1960s so we'll have to pull dusty history textbooks from shelves to settle the question about the (lack of) importance of advertising them in 1912. | |
| Dec 10, 2022 at 2:34 | comment | added | Timothy Chow | @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. | |
| Dec 10, 2022 at 2:23 | comment | added | fedja | @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. | |
| Dec 10, 2022 at 1:45 | comment | added | Joseph O'Rourke | @TimothyChow: Excellent point re unattackable vs. research-spurring problems. | |
| Dec 10, 2022 at 1:27 | comment | added | Will Jagy | 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 | |
| Dec 10, 2022 at 1:27 | comment | added | Timothy Chow | 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. | |
| Dec 10, 2022 at 1:01 | history | asked | Joseph O'Rourke | CC BY-SA 4.0 |