Timeline for Why do we need to represent integers as the sum of three cubes? [closed]
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Dec 2, 2021 at 19:26 | vote | accept | zeraoulia rafik | ||
| Dec 2, 2021 at 15:59 | comment | added | Timothy Chow | Most mathematics is either trivial or hopelessly difficult. All the action lies in the tiny sliver of territory in between. The sum of three cubes lives in that sliver. We have a good grasp of quadratic equations, and a decent grasp of elliptic curves. The sum of three cubes is the next step up. Based on known technology, we have no reason to believe that $a^3 + b^3 + c^3 = k$ is unsolvable for $k\not\equiv \pm 4\pmod 9$, but we also can't prove it has solutions. The hope is that studying this equation will lead to new techniques, either of finding solutions or of proving unsolvability. | |
| Dec 2, 2021 at 5:16 | history | closed |
Wojowu LSpice David Roberts♦ Steven Landsburg Stopple |
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| Dec 2, 2021 at 4:47 | comment | added | BPP | @NickS I'm skeptical of that claim. Certainly there are a few notable pure math problems that stimulated the development of "new results and tools [...] with applications to many other problems", but I'm not convinced that the phenomenon occurs "very often". Do you know of any empirical data that sheds light on how often that happens? In any case, the development of new results and tools is not the primary reason we study problems like this. We study them because they're interesting, and we would be happy to see solutions to these problems even if they didn't lead to new results and tools. | |
| Dec 2, 2021 at 4:11 | comment | added | user164898 | These questions come up naturally. It is not just that their value comes from the methods used to solve them. I am not a number theorist, but occasionally I have been in situations where, for example, I run a spectral sequence of modules over the p-adic integers, and a certain family of differentials hit elements of the form a^2 + b^2 in various bidegrees. Consequently, to count the orders of the groups surviving the spectral sequence, you need to know which powers of p are equal to sums of 2 squares. "Which numbers are sums of m nth powers?" is a useful type of question to be able to answer. | |
| Dec 2, 2021 at 1:50 | comment | added | Nick S | Very often trying to answer question like this leads to new results and tools (and some times to new areas) with applications to many other problems. Sometimes is the trip that is fascinating, not the destination. | |
| Dec 2, 2021 at 1:49 | answer | added | David Roberts♦ | timeline score: 11 | |
| Dec 2, 2021 at 0:30 | comment | added | LSpice | Questions worded like this always rub me the wrong way, though I understand and appreciate the instinct behind them. I think the same question could be asked in a much less provocative way—even just saying "What are the implications of writing a number as a sum of three cubes?" could explore this territory without devaluing the work already put into it. (For example, does it matter that a specific number can or cannot be written as a sum of 3 cubes, or only whether or not all numbers can be so written? I don't know, and I think that would be an interesting question!) | |
| Dec 2, 2021 at 0:27 | history | edited | LSpice | CC BY-SA 4.0 |
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| Dec 2, 2021 at 0:20 | review | Close votes | |||
| Dec 2, 2021 at 5:26 | |||||
| Dec 2, 2021 at 0:00 | comment | added | Wojowu | "is it just curiosity or will it provide a new addition to number theory?" It would be both! The very fact that 114 is representable would be a new addition to number theory, although that alone is merely a curiosity. What we are after is general methods which could tackle this problem in some generality and not just specific numbers. | |
| Dec 1, 2021 at 23:58 | history | edited | zeraoulia rafik | CC BY-SA 4.0 |
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| Dec 1, 2021 at 23:56 | comment | added | Wojowu | Why not? We have considered many different problems of the sort "which integers can be represented in some particular form", and this is arguably the simplest one which is still open. | |
| Dec 1, 2021 at 23:44 | history | asked | zeraoulia rafik | CC BY-SA 4.0 |