Timeline for Bounds on the size of rough numbers
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 24, 2019 at 22:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 27, 2019 at 21:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Mar 29, 2019 at 20:22 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Feb 28, 2019 at 11:29 | comment | added | Simd | @WlodAA An answer incorporating a major breakthrough in number theory would also be acceptable. | |
| Feb 27, 2019 at 20:53 | comment | added | Wlod AA | What's wrong with "a major breakthrough in number theory"? | |
| Feb 27, 2019 at 17:48 | answer | added | Gerhard Paseman | timeline score: 1 | |
| Feb 17, 2019 at 16:54 | comment | added | Jan-Christoph Schlage-Puchta | @Gerhard Paseman: The two ranges actually overlap. To get explicit results I only use Chebychev type inequalities, not the prime number theorem. Looking only at primes we find that the claim is true if $\pi(Bk)>k+B$, which is true for $B>2\log k$. On the other hand the first approach works for $k>3^{B/\log B}\log B$. Thus the claim is true for all $B<30$ and all $k<5\cdot 10^8$. Counting the number of 30-rough integers below $5\cdot 10^8$ greatly reduces these bounds. Then check the small cases individually. | |
| Feb 11, 2019 at 17:36 | comment | added | Gerhard Paseman | One can also use existing estimates on $\pi(2N)-\pi(N)$ for $N=B,2B, 4B$, etc. to bump up $k$. However I do not know how large a value of $k$ you can achieve with this. Gerhard "Less Sure About Middling K" Paseman, 2019.02.11. | |
| Feb 11, 2019 at 17:09 | comment | added | Gerhard Paseman | For k larger than something like $2^{\pi(B)}\log B$ this is clear from counting numbers coprime to the primes at most $B$. For better estimates one can use bounds on Jacobsthal's function to show it holds for k greater than g(P), where P is the product of the $m$ many primes at most P and g(P) is bounded above by $m^{3+3.81\log\log m}$ (and likely also by $m^2$). Gerhard "Don't Know About Small K" Paseman, 2019.02.11. | |
| Feb 11, 2019 at 16:50 | history | asked | Simd | CC BY-SA 4.0 |