Timeline for Asymptotics for primality of sum of three consecutive primes
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2011 at 17:45 | comment | added | joriki | @Noam: Yes, sorry, $1/4$. Regarding you other questions, see above. Yes, seeing the ratio converge to $2.30$ isn't feasible, I think; what might be feasible, though, is to model these effects well enough to support the conclusion that they eventually disappear. | |
| Aug 10, 2011 at 17:44 | history | edited | joriki | CC BY-SA 3.0 |
added data on divisibility and residue distribution; added 14 characters in body
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| Aug 10, 2011 at 15:42 | comment | added | Noam D. Elkies | @joriki: I guess you mean $0.2$ instead of $1/4$, not $3/4$. Also -- how close are $l=5$ and $l=7$, and does there seem to be significant dependence between $l=3$ and $l=5$? Presumably all these effects eventually disappear but it looks like it would require lots of computation to see the ratio converge to 2.30. | |
| Aug 10, 2011 at 15:22 | comment | added | joriki | @Noam: Yes, I made the same observations and was just doing some experiments on that -- the approach is quite slow; even at the end of the table the probability for $l=3$ is only about $0.2$ instead of the asymptotic $3/4$; this is due to repeated residues being significantly less likely than alternating residues, and this effect only decays slowly as the gaps between the primes become wider. | |
| Aug 10, 2011 at 14:07 | comment | added | Noam D. Elkies | See my comment on Álvaro Lozano-Robledo's computation. What happens when $n / \log n$ is replaced by the better estimate $\sum_{m=1}^n 1/\log(3p_m)$? Up to $n=10^5$ this made the observed constant much larger than $\lambda$ (about $2.8$). It seems that for the smallest odd primes $l$ it takes a while for the probability of $l \mid p_n + p_{n+1} + p_{n+2}$ to approach the expected value $(l^2-3l+3)/(l-1)^2$. | |
| Aug 10, 2011 at 7:17 | history | undeleted | joriki | ||
| Aug 10, 2011 at 7:17 | history | edited | joriki | CC BY-SA 3.0 |
improved conclusion
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| Aug 10, 2011 at 7:06 | history | deleted | joriki | ||
| Aug 10, 2011 at 6:26 | history | edited | joriki | CC BY-SA 3.0 |
more sensible rounding
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| Aug 10, 2011 at 6:17 | history | answered | joriki | CC BY-SA 3.0 |