Timeline for Existence of an explosive prime
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2022 at 10:39 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
motivation highlighted
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| Feb 21, 2022 at 11:35 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
Shorter definition of the map f (which does not use rad)
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| Feb 21, 2022 at 11:16 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
deleted 28 characters in body
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| Feb 21, 2022 at 11:14 | comment | added | Sebastien Palcoux | @ChrisWuthrich: You are right, square-free part is confusing (I will remove that), I just mean the radical as defined on this link, i.e. the biggest square-free divisor. | |
| Feb 21, 2022 at 10:47 | comment | added | Chris Wuthrich | I think $\operatorname{rad}(p^2)=p$ for a prime, so it is not the square-free part. Which do you mean? | |
| Feb 21, 2022 at 10:10 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
new data
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| Feb 15, 2022 at 8:27 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
experimental data to support density estimate
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| Feb 15, 2022 at 7:53 | comment | added | Sebastien Palcoux | @PeterTaylor: in my computation I iterated the map $f$ to get at least twelve new primes (except if the sequence becomes constant, of course). Even then, nothing guarantee explosion (otherwise that would provide a proof to Carmichael's totient conjecture). We can just say that the initial prime looks explosive. | |
| Feb 15, 2022 at 7:16 | comment | added | Peter Taylor | Perhaps it would be clearer as $\{dp + 1 | d \textrm{ is a factor of } 1806\}$. Any initial $p$ will accumulate primes $2, 3, 7, 43$ merely by being a multiple of $1$, but the expected number of further primes accumulated is $O(\frac1{\log p})$ (and picking up one more prime doesn't guarantee explosion). | |
| Feb 15, 2022 at 5:27 | comment | added | Sebastien Palcoux | Typos: $(407, 428)$ should be $(428,407)$, and $(451,419)$ should be $(419,451)$. | |
| Feb 15, 2022 at 2:29 | comment | added | Sebastien Palcoux | @PeterTaylor Sorry my use of "heuristically" is no proper, I should say "experimentally": I counted $342$ primes looking explosive between $10^{24}$ and $10^{24} + 10^6$, and $259$ ones between $10^{25}$ and $10^{25} + 10^6$, whereas this heuristic with $c=1$ predicts $(315,291)$. FYI, I counted $(407, 428)$ Sophie Germain primes, and $(451,419)$ twin primes, in these intervals, whereas their heuristic (with $c=1.32032\dots$) predicts $(426,384)$. About your comment, can you elaborate? $1806 = 2^13^17^143$, ok but $6$ is a factor of $1806p$ (for all $p$) whereas $7$ is not explosive. | |
| Feb 14, 2022 at 15:38 | comment | added | Sylvain JULIEN | @SébastienPalcoux : I vainly tried to find one, unfortunately I don't remember on which website (MO or MSE) I asked the considered question nor its title. | |
| Feb 14, 2022 at 13:01 | comment | added | Peter Taylor | Heuristically the number of primes $p < n$ for which $u_5(p) > 5$ is also $O(\frac{n}{\log^2 n})$. Does your heuristic assume that if the process gets bootstrapped by a prime in $\{d + 1 | d \textrm{ is a factor of } 1806p\}$ then it will be explosive? | |
| Feb 14, 2022 at 9:58 | comment | added | Sebastien Palcoux | @SylvainJULIEN: can you put a link to what you are talking about? | |
| Feb 14, 2022 at 9:57 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
heuristic density as for twin or sophie germain primes
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| Feb 13, 2022 at 15:32 | comment | added | Sylvain JULIEN | That reminds me a bit my investigations about the map $n\mapsto nr_{0}(n)$, whose the sequence of iterates gets constant if some term thereof is half the sum of twin primes, perhaps the same kind of tools could be used in both problems. | |
| Feb 13, 2022 at 7:11 | comment | added | Sebastien Palcoux | en.wikipedia.org/wiki/168_(number) | |
| Feb 13, 2022 at 5:58 | comment | added | Gerry Myerson | I always thought of 168 as the place where you could transfer between the 1 and the A. en.wikipedia.org/wiki/… | |
| Feb 12, 2022 at 16:32 | comment | added | Roland Bacher | I always thought of 168 as of the cardinal of the second smallest simple group. Nice to learn that it is also the number of primes smaller than 1000. | |
| Feb 12, 2022 at 9:29 | history | edited | Sebastien Palcoux | CC BY-SA 4.0 |
minor edit
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| Feb 12, 2022 at 8:28 | history | asked | Sebastien Palcoux | CC BY-SA 4.0 |