Timeline for Are there primes of every Hamming weight?
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2023 at 0:40 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Jan 27, 2023 at 22:48 | comment | added | Gerry Myerson | @Wlod, smallest prime with Hamming weight $n$ is tabulated at oeis.org/A061712 and begins $2,3,7,23,31,311,127,\dots$ (note failure of monotonicity). Some references to the literature are there. | |
| Jan 27, 2023 at 12:55 | comment | added | Wlod AA | Actually, let $\ H^\searrow(n)\ $ be the smallest prime of Hamming weight $\ n.\ $ Indeed, the largest prime $\ H^{\nearrow}(n)\ $ that has Hamming weight $\ n\ $ presents a true challenge too. Furthermore, the relation between these two sequences is far from obvious! | |
| Jan 27, 2023 at 11:42 | comment | added | Wlod AA | For instance: $\ H(1)=2,\ H(2)=3,\ H(3)=7,\ \ldots$ | |
| Jan 27, 2023 at 11:38 | comment | added | Wlod AA | In view of the @fedja answer, the question can be reformulated in a numeric way: compute the Hamming primes $\ H(n),\ $ where prime $\ H(m)\ $ is the smallest among all primes that have Hamming weight equal to n. | |
| Jan 27, 2023 at 9:50 | history | edited | YCor | CC BY-SA 4.0 |
formatting, copied question to body
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| Jan 27, 2023 at 8:38 | comment | added | Roland Bacher | Is there a reason not to believe that the number of $1$'s in primes of size roughly $2^n$ should asymptotically be given by a suitably rescaled normal distribution centered at $n/2$? | |
| S Mar 15, 2018 at 1:35 | history | suggested | Jeppe Stig Nielsen | CC BY-SA 3.0 |
fix OEIS links
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| Mar 14, 2018 at 23:35 | review | Suggested edits | |||
| S Mar 15, 2018 at 1:35 | |||||
| Apr 27, 2010 at 20:12 | answer | added | Junkie | timeline score: 4 | |
| Apr 27, 2010 at 17:59 | vote | accept | dakota | ||
| Apr 27, 2010 at 16:07 | answer | added | Felipe Voloch | timeline score: 2 | |
| Apr 27, 2010 at 16:01 | answer | added | Ben Green | timeline score: 56 | |
| Apr 27, 2010 at 15:18 | comment | added | dakota | fedja, let me see if I understand this correctly. The paper (dmg.tuwien.ac.at/drmota/DMRcomp2.pdf) gives (5) for the number of primes, p, less than a given x whose Hamming weight is near half the length of p. The expression is (something positive)*(something unbounded)*(something greater than 1) and is therefore ≥1 for large x. I haven't looked at it long enough, but I don't immediately see why it is the case that we can specify a length here and find a prime of that length. It looks like you can do this in (4), but how do you show for any k we have an x so (4) is ≥1? | |
| Apr 27, 2010 at 14:28 | answer | added | ogerard | timeline score: 0 | |
| Apr 27, 2010 at 3:57 | comment | added | Qiaochu Yuan | fedja, you should post that reference as an answer! | |
| Apr 26, 2010 at 22:49 | comment | added | fedja | See the reference mentioned in artofproblemsolving.com/Forum/viewtopic.php?f=56&t=246494 This doesn't resolve the question for all $n$, just for sufficiently large $n$, of course, but I do not think that the existence of a few exceptional values (which is highly unlikely) will make a lot of difference for anything. | |
| Apr 26, 2010 at 21:04 | comment | added | Thomas Bloom | (A very weak version assuming a difficult conjecture) It would follow from the existence of arbitrarily long Cunnigham Chains (en.wikipedia.org/wiki/Cunningham_chain) that there are arbitrarily long strings of consecutive integers n,n+1,...,n+k such that they are all the Hamming numbers of primes. Based on the links with this, and difficult conjectures like the Fermat Primes and Schinzel's hypothesis, I agree with Kevin that this looks like an open problem. | |
| Apr 26, 2010 at 19:28 | comment | added | dakota | @Joel Quite so. Edited to reflect this. | |
| Apr 26, 2010 at 19:27 | history | edited | dakota | CC BY-SA 2.5 |
new links, edit suggested by jdh
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| Apr 26, 2010 at 19:23 | comment | added | Kevin Buzzard | This will surely be an open problem. | |
| Apr 26, 2010 at 18:39 | comment | added | Joel David Hamkins | I think you want positive integer n, since the negative integers have no chance, and n=0 also does not occur. | |
| Apr 26, 2010 at 18:32 | comment | added | Joel David Hamkins | Welcome to MO, dakota. | |
| Apr 26, 2010 at 18:29 | history | asked | dakota | CC BY-SA 2.5 |