Timeline for Asymptotic behavior in a modular color-cycling problem
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 15 at 21:08 | vote | accept | PianothShaveck | ||
| Aug 15 at 11:25 | comment | added | PianothShaveck | More extensive (but not really rigorous) search found a lot more properties that hold up to $10^8$, and drastically improved the bounds so they're now 38.72488% and 49.4805% respectively. I've updated the main post showing the properties that were found. However, the patterns that hold for blue lights seem pretty much random. | |
| Aug 15 at 10:19 | comment | added | 1001 | I added some code | |
| Aug 15 at 10:19 | history | edited | 1001 | CC BY-SA 4.0 |
added 1460 characters in body
|
| Aug 14 at 23:21 | comment | added | PianothShaveck | With the help of computer search the best I was able to get is a lower bound of $\frac{14762}{59049} \approx 24.9996$ % and an upperbound of $\frac{29525}{59049} \approx 50.0008$ %. Aside searching for powers of 3, I also searched for the congruence classes of all the primes up to 104729. There doesn't appear to be any other pattern. The only congruence classes that represent the blue lights appear to be in the form $2 * 3^n$. However, improving this result further only appears to get closer and closer to 25% and 50% respectively. | |
| Aug 14 at 19:28 | comment | added | 1001 | My guess is that it is possible to prove convergence following the two steps outlined in the post (but I haven't checked, so I might be wrong). However that would not give the constant it converges to. I'm not sure the constant which it converges to could be determined analytically since the distribution of small prime factors mod 3 seems to matter a lot. However, I don't know much about number theory, so maybe someone else has some intuition about this? On the other hand, stronger upper and lower bounds could be found by checking congruence classes computationally. | |
| Aug 14 at 18:24 | comment | added | PianothShaveck | Second fraction was supposed to be $\frac{365}{729}$. | |
| Aug 14 at 18:14 | comment | added | PianothShaveck | With more search I'm able to bound them between $\frac{182}{729}$ and $\frac{365}{364}$. Basically between 25% and 50%. | |
| Aug 14 at 17:53 | comment | added | PianothShaveck | A bit of an improvement over what you discovered: if $n \equiv 5 \bmod 15$, then the light is blue. If $n \equiv 6 \bmod 18$, that's also blue. In $\mod 30$, we can say that ${1, 4, 7, 10, 13, 16, 19, 22, 25, 28} \bmod 30$ are always red, and $\{2, 5, 8, 14, 20, 26\} \bmod 30$ are always blue. This bounds the fraction of blue lights between $\frac{1}{5}$ and $\frac{2}{3}$. The result can possibly be improved further but it might take some time. | |
| Aug 14 at 16:23 | comment | added | PianothShaveck | This explanation helps a lot, especially the lower bound on blue lights. Do you think a more formal proof could establish the exact asymptotic behavior of the blue lights as $k$ increases? Any suggestions on how to approach that? | |
| Aug 14 at 10:27 | history | edited | 1001 | CC BY-SA 4.0 |
added 464 characters in body
|
| Aug 14 at 9:59 | history | answered | 1001 | CC BY-SA 4.0 |