Timeline for Covering the primes by arithmetic progressions
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 21, 2010 at 23:14 | comment | added | Daniel Litt | @Joseph: Ah oops. | |
| Nov 21, 2010 at 20:46 | vote | accept | Joseph O'Rourke | ||
| Nov 21, 2010 at 20:46 | history | edited | Joseph O'Rourke | CC BY-SA 2.5 |
Addendum upon accepting an answer. Adding a tag.
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| Nov 21, 2010 at 3:39 | comment | added | Gerry Myerson | @Joseph, I missed that 19. | |
| Nov 21, 2010 at 1:29 | comment | added | Joseph O'Rourke | @Gerry: I did mean cover and not partition, as my second example illustrated (19 is repeated). But I see and appreciate your point. | |
| Nov 21, 2010 at 0:38 | comment | added | Gerry Myerson | The use of the word, "exactly", together with the examples given, suggests to me that you want disjoint progressions. If that's the case, then I don't think it's quite equivalent to the question about primes belonging to progressions. E.g., if $p\lt q\lt r$ are prime, and the only 3-term progression containing $q$ is $p,q,2q-p$, and the only 3-term progression containing $r$ is $p,r,2r-p$, then you're sunk, no? | |
| Nov 20, 2010 at 23:31 | answer | added | George Lowther | timeline score: 10 | |
| Nov 20, 2010 at 21:33 | comment | added | George Lowther | @Daniel: How about d=3. Is that open? | |
| Nov 20, 2010 at 21:29 | comment | added | Joseph O'Rourke | @Fedor, Daniel: Oh, I see, sorry for misunderstanding. | |
| Nov 20, 2010 at 21:21 | comment | added | Daniel Litt | As Fedor suggests, the question "does every prime belong to an arithmetic progression of primes of length $d$," to which this question is equivalent, is very, very open for large $d$. So this question should probably be re-tagged "open-question"; that said this statistic is probably an interesting one for many other subsets of the integers. | |
| Nov 20, 2010 at 20:49 | comment | added | Fedor Petrov | it is not a result, but rather widely open conjecture... | |
| Nov 20, 2010 at 20:15 | comment | added | Joseph O'Rourke | @Fedor: Your comment seems to definitively answer my question: $L_{\max}$ has no upper bound! I did not know the existence result you use. Thanks! | |
| Nov 20, 2010 at 20:02 | comment | added | Fedor Petrov | some general heuristics and explicit hard conjectures suggest that you may cover primes by (even disjoint) arithmetic progressions of arbitrary length $n$. Indeed, if you have already covered all primes up to $p$ (but not $p$), then there must exist an arithmetic progression $\{\p,p+d,p+2d,\dots,p+(n-1)d}$ for infinitely many $d$'s. So cover $p$ and proceed. | |
| Nov 20, 2010 at 19:50 | history | asked | Joseph O'Rourke | CC BY-SA 2.5 |