Timeline for Nonlinear equations in integers
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| May 23, 2012 at 4:03 | answer | added | Mark Lewko | timeline score: 2 | |
| May 23, 2012 at 2:24 | answer | added | user22202 | timeline score: 0 | |
| May 22, 2012 at 16:46 | comment | added | user9072 | @Ben Green: Thank you for the correction! Not sure how I could miss that in the first place. | |
| May 22, 2012 at 16:34 | comment | added | Ben Green | In fact there is no set of size $o(N)$ which misses all triples $(3k, 4k, 5k)$ by the following type of argument: consider such triples with $|k - N/10| < N/100$ (say). These are all disjoint. Therefore any set consisting of $98\%$ of ${1,..,N}$ contains a pythagorian triple. Computation of the exact density might be tricky. | |
| May 22, 2012 at 16:30 | comment | added | Ben Green | One of the references here, I bet: algo.inria.fr/csolve/triple | |
| May 22, 2012 at 16:28 | comment | added | Ben Green | I'm pretty sure, actually, that all sets of $99\%$ of $[1,...,N]$ contain a triple $(3k, 4k, 5k)$. I'm also pretty sure I can find a reference for this fact given a minute or two. | |
| May 22, 2012 at 16:21 | comment | added | Ben Green | Quid: what about $3^2 + 4^2 = 5^2$? | |
| May 22, 2012 at 15:44 | comment | added | user9072 | Inspection modulo 12 suggests that 5/6 (all not divisible by 6) is also doable. | |
| May 22, 2012 at 15:33 | history | edited | Siming Tu | CC BY-SA 3.0 |
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| May 22, 2012 at 15:29 | comment | added | Ben Green | Siming, I think that taking the squares $x^2$ with $x$ either odd or $\equiv 2 \mod{4}$ gives you a set consisting of $\frac{3}{4}$ of the squares with no solution to $x^2 + y^2 = z^2$ (look mod $8$). Probably it's true that if you take $99 \%$ of the squares then there's a solution to $x^2 + y^2 = z^2$. I don't immediately have a feel for whether this is doable or not; I'll get back to you. Certainly use of the circle method will be problematic if one proceeds naively. | |
| May 22, 2012 at 15:07 | answer | added | user9072 | timeline score: 4 | |
| May 22, 2012 at 14:52 | comment | added | Siming Tu | @Mark Sapir: I'm sorry that I did not think much when I gave examples. For the equation $x^2+y^2=z^2$, there is a set of density 1/2 such that the equation does not have any solution in it. However, we can still ask how large the density can be, I add a note in my question. | |
| May 22, 2012 at 14:50 | history | edited | Siming Tu | CC BY-SA 3.0 |
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| May 22, 2012 at 14:42 | history | edited | Siming Tu | CC BY-SA 3.0 |
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| May 22, 2012 at 14:27 | comment | added | user6976 | @Siming: Why don't you think before you add examples of equations? Ben Green gave a non-trivial example before. | |
| May 22, 2012 at 14:18 | history | edited | Siming Tu | CC BY-SA 3.0 |
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| May 22, 2012 at 14:15 | answer | added | Ben Green | timeline score: 6 | |
| May 22, 2012 at 14:02 | answer | added | user6976 | timeline score: 3 | |
| May 22, 2012 at 13:56 | history | asked | Siming Tu | CC BY-SA 3.0 |