Timeline for Low-prime intervals
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7 at 3:42 | comment | added | Wlod AA | I instantly upvoted @TheBestMagician comment, thus, let me mention -- using a Computer Science metaphor -- my q. was written in a higher-level language while their comment used assembler. | |
| Jun 7 at 3:17 | comment | added | Wlod AA | @DaveBenson "completely" -- actually, this opens a (positive!) Pandora's box. | |
| Jun 7 at 1:02 | answer | added | so-called friend Don | timeline score: 8 | |
| Jun 6 at 22:35 | comment | added | Dave Benson | @so-calledfriendDon That seems to settle the question completely! I'm surprised that it's "reasonably elementary" as you put it. | |
| Jun 6 at 21:48 | history | edited | Wlod AA | CC BY-SA 4.0 |
word "be" was missing
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| Jun 6 at 21:18 | comment | added | Wlod AA | Oh, no, how could I forget the so familiar case of $\ M(48\,\ 50) = \frac 12\ $ -- thank you, Dave. | |
| Jun 6 at 20:59 | comment | added | so-called friend Don | Someone should mention the paper ON STRINGS OF CONSECUTIVE INTEGERS WITH NO LARGE PRIME FACTORS, by Balog and Wooley. (One of their results: For each positive integer $k$ and each $\epsilon > 0$, there are infinitely many $n$ where none of $n+1, \dots, n+ k$ has a prime factor exceeding $n^\epsilon$.) The proof is reasonably elementary. | |
| Jun 6 at 17:15 | comment | added | Dave Benson | Sloane's A355434 gives the beginnings of such sequences as follows: 1, 8, 48, 1518, 5828, 28032, 304260, 290783, 1255500, 4325170, 11135837, 18567909, 321903029, 1394350275, 287946949, 1659945758, 38882519234. So there are such sequences at least up to length 17. It doesn't indicate whether this goes on for ever, but I suspect it does. | |
| Jun 6 at 14:07 | history | edited | Wlod AA | CC BY-SA 4.0 |
a typo
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| Jun 6 at 13:45 | history | edited | Wlod AA | CC BY-SA 4.0 |
one more example
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| Jun 6 at 13:31 | comment | added | Wlod AA | @DaveBenson, thank you for your examples. In my comment, I just wanted things to get rolling, I was not attempting a most ambitious conjecture. ******* Dave, you're very welcome to post your examples as, let people see partial progress. ******** In the light of your examples, the conjecture about arbitrary long $\ [a\,\ b]\ $ such that $\ M(a\,\ b)<\frac 12 $ is worth stating (it may go either way :) ). | |
| Jun 6 at 12:52 | comment | added | Dave Benson | I suppose the point of my examples was that your question in the comments about $M(n-1\ n+1)\leqslant \frac{1}{2}$ was not nearly bold enough. My last example was a sequence of eight consecutive integers with no prime divisors as large as the square root. I tried to find nine, but my computer got bored and started humming tunes from the Marriage of Figaro at me. | |
| Jun 6 at 11:23 | comment | added | Stanley Yao Xiao | Is it fair to interpret the question as asking for the behaviour of functions $a(x)$ such that for infinitely many $x$ there are intervals of the shape $[x - a(x), x]$ which consist of only $x^f$ smooth integers, for some $f \in (0,1)$? | |
| Jun 6 at 10:51 | comment | added | Dave Benson | 290783*290784*290785*290786*290787*290788*290789*290790 only involves primes up to 523. | |
| Jun 6 at 10:40 | comment | added | Dave Benson | 28032*28033*28034*28035*28036*28037 only involves primes up to 163. | |
| Jun 6 at 10:38 | comment | added | Dave Benson | 5828*5829*5830*5831*5832 only involves primes up to 67. | |
| Jun 6 at 10:30 | comment | added | Dave Benson | 17575*17576*17577*17578 only involves primes up to 47. | |
| Jun 6 at 10:18 | comment | added | Dave Benson | 2430*2431*2432 only involves primes up to 19, and 13310*133111*13312 only involves primes up to 29. | |
| Jun 6 at 9:20 | history | edited | Wlod AA | CC BY-SA 4.0 |
a funny mth typo
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| Jun 6 at 5:50 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting and typo fixing
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| Jun 6 at 2:50 | comment | added | Wlod AA | My note is significantly related to a "Problem of Stormer" -- see paper "On a problem of Stormer" (1962) by D.H.Lehmer. | |
| Jun 6 at 2:42 | comment | added | Wlod AA | There are also a lot of detailed question. E.g. Is there a natural number $\ n>1\ $ such that $\ M(n-1\ \ n+1)\ \le \frac12\ ?$ | |
| Jun 6 at 2:38 | history | edited | Wlod AA | CC BY-SA 4.0 |
a typo
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| Jun 6 at 0:55 | comment | added | TheBestMagician | Well the maximum term occurs when $p$ is maximized, so the question is if there are long intervals $[a,b]$ such that the largest prime divisor of $\frac{b!}{(a-1)!}$ is $\le b^f$. | |
| Jun 6 at 0:40 | review | Close votes | |||
| Jun 10 at 3:05 | |||||
| Jun 6 at 0:00 | history | edited | Wlod AA | CC BY-SA 4.0 |
F_N was missing
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| Jun 5 at 23:38 | history | asked | Wlod AA | CC BY-SA 4.0 |