Timeline for Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10, 2019 at 12:05 | comment | added | Zhi-Wei Sun | I don't want to argue with anyone here. Different people have different viewpoints. For example, if Goldbach's conjecture only holds for even numbers greater than $10^{100}$, it would not be so famous. Usually people like concise results or conjectures. | |
| Mar 10, 2019 at 9:33 | comment | added | Alex B. | "I'm only interested in writing every (not sufficiently large) $n\in\mathbb{N}$ in a desired form". But why? That is the question that several people, including myself, would like to know the answer to. What would it teach you about numbers if you knew that some particular a priori obviously fairly dense set contains all integers, rather than all with, say, one exception? | |
| Mar 10, 2019 at 1:47 | comment | added | Zhi-Wei Sun | For 2-4-6-10, the first counterexample is 68286. Unlike analytic number theorists, I'm only interested in writing every (not sufficiently large) $n\in\mathbb N$ in a desired form. | |
| Mar 10, 2019 at 1:43 | comment | added | Lucia | Just take $1/2+1/4+1/6+1/10$ which is a tiny bit more than $1$, and I'm sure you can form some conjecture for large $n$. | |
| Mar 10, 2019 at 1:39 | comment | added | Zhi-Wei Sun | This problem is particular since $1/2+1/4+1/6+1/8\approx 1.04$. I don't know any conjecture representing each $n\in\mathbb N$ as $\sum_{i=1}^k x_i^{a_i}$ with $\sum_{i=1}^k1/a_i<1.04$ | |
| Mar 10, 2019 at 1:37 | comment | added | Lucia | I'm holding this problem to a high standard because the OP is a professional mathematician, and should be aware of the background for such problems. There are of course a lot of amateurs posing problems on MO. But from a professional mathematician, I imagine that there is some particular reason that you find this interesting, and I would like to understand what that is. I am not intending to be offensive, and of course you should feel free to ignore my answer or my comments. | |
| Mar 10, 2019 at 1:35 | comment | added | Lucia | Yes, indeed. There are already well established challenging problems to innovate upon. E.g. in this case Waring's problem, and then (with a significant drop in interest) the Waring problem for unlike powers etc. I ask again, what do you think is new that we learn from this problem, which we did not know from the circle method? Why might one expect this to stimulate work, compared to the many problems that workers in the circle method are already familiar with? | |
| Mar 10, 2019 at 1:32 | comment | added | Zhi-Wei Sun | New methods are usually stimulated by challeging problems which could not be solved by the known methods. There are many interesting open problems which look hopeless to be solved, such as GRH, Schinzel's Hypothesis and whether there are infinitely many Mersenne primes. | |
| Mar 10, 2019 at 1:25 | comment | added | Lucia | I'm not sure you've read through my answer. I point out clearly why the circle method indeed will not give that. Exactly the point for your conjecture too. Do you really feel that your conjecture is so different from the Waring problem for unlike powers? | |
| Mar 10, 2019 at 1:24 | comment | added | Zhi-Wei Sun | Number theorists are still interested in the exact value of $G(3)$ in Waring's problem though the circle method even does not work for $G(3)\le6$. | |
| Mar 10, 2019 at 1:19 | comment | added | Lucia | And my point is that you could promise a whole lot more, and it would still be hopeless! Why not 246810 dollars? And of course I would expect you to be aware of the circle method -- although the problem gave no indication of this. Sorry, but conjectures of this type are routine, and you should at least flesh out the main term before stating such a conjecture. | |
| Mar 10, 2019 at 1:18 | comment | added | Zhi-Wei Sun | I know the limitation of the circle method. If the 2-4-6-8 conjecture could be easily proved by the current circle method without creative innovation, I would not promise $2468 prize for its proof. | |
| Mar 9, 2019 at 22:19 | comment | added | Lucia | @Wojowu: No; my point is that one needs the sum to be $>2$ in order to have any hope! In each situation, what is known would vary depending on our knowledge of the relevant exponential sums. | |
| Mar 9, 2019 at 22:18 | comment | added | Wojowu | Is there some general kind of result which says that if $\sum 1/a_k>2$, then sufficiently large integers are of the form $\sum x_k^{a_k}$? I'm afraid I am not well-versed enough in additive number theory to see whether the argument generalizes. | |
| Mar 9, 2019 at 21:53 | history | answered | Lucia | CC BY-SA 4.0 |