|WikiProject Mathematics||(Rated Start-class, Low-priority)|
How do you prove a Sierpinski number?
Why was it possible to rigorously prove (by induction, perhaps?) that 78557 is Sierpinski, while "the Seventeen" are indeterminate and only testable by brute force? If one of the remaining eight has an extreme n value (let's say > 2^64), the speed of light dictates that our current forms of computer hardware will probably never search that high. Frankie 21:59, 21 December 2005 (UTC)
- To show that some k is a Sierpinski number requires proving the compositeness of an infinite number of k2n+1 values, and thus requires some mathematical finesse. The basic proof involves breaking the various possibilities for n down into cases and showing that all cases have factors. For example, 78557·2n+1 ≡ 0 (mod 3) for all odd n.
- However, to show that k is not a Sierpinski number it suffices to simply present a counterexample, and a counterexample can be searched for using brute force. Mathematical finesse disposed of most of the k<78557, but the last few proved difficult, so the holdouts are being attacked by simple brute force. 184.108.40.206 06:28, 27 February 2007 (UTC)
Naming: Sierpiński versus Sierpinski
Apparently this is named after Wacław Sierpiński, so why doesn't the spelling reflect that accurately here, as it does for Sierpiński's constant and Sierpiński curve? Does this need fixing up? —DIV 220.127.116.11 03:40, 6 January 2007 (UTC)
- None of the first 100 Google hits on "Sierpiński number" -wikipedia say "Sierpiński". Google scholar is apparently more character sensitive here and gives 0 hits on "Sierpiński number" but 10 on "Sierpinski number". If "Sierpinski number" dominates completely then I think we should stay with it, even if somebody should find a reference saying Sierpiński number which i just created a redirect on. I haven't examined other articles named after Sierpiński. PrimeHunter 12:08, 6 January 2007 (UTC)
- Okay, it is easy to play with statistics. Altavista returns the following:
- "Waclaw Sierpinski" - 2140
- "Wacław Sierpiński" - 829
- "Wacław Sierpinski" - 40
- "Waclaw Sierpiński" - 18
- So according to the majority, "Waclaw Sierpinski" is the way to go ...even though it is not correct. Yes, many contributors to Wikipedia seek comfort in Google's results, but it would be nice to settle these questions by choosing the 'correct' option instead, with all due respect.
- My philosophy would be the converse of yours: create the page where it should be, and redirect people who don't know better or (just as likely) can't conveniently enter the correct letters. If you can see the logic in keeping the entry at Wacław Sierpiński, then I think you should be able to see the logic for the derivative articles.
- Regards, DIV 18.104.22.168 09:07, 25 January 2007 (UTC)
- Wacław Sierpiński is the real name of a real person, and I think it should be used. But I think a technical term should usually be under the most common name for that particular thing, no matter whether the common name is considered 'correct' by some other rule, e.g. who it is named after, or who it deserves to be named after, or who first studied it. See Wikipedia:Naming conventions (common names). I didn't find a single source saying "Sierpiński number" and lots saying "Sierpinski number". See also Wikipedia:Naming conventions (use English)#Disputed issues (the term "Sierpinski number" may be an English invention and not a native name in Wacław Sierpiński's language) and Wikipedia:Naming conventions (standard letters with diacritics). PrimeHunter 15:32, 25 January 2007 (UTC)
- That is all very interesting. I too found no websites referring to "Sierpiński number" (although it should be noted that this page uses the term, but was not found!), and a number referring to "Sierpinski number". What was more interesting, however, is that ALL of those sites that referred to the "Sierpinski number" also incorrectly referred to the man himself as Sierpinski. ...Except for Wikipedia and the sites echoing it. So I would contend that if they can't get the person's name right, then they are not good references for the mathematical terms either.
- I have tried to search the scholarly literature, but haven't managed to find an appropriate search engine on such a database (similar to what you described earlier). ...This strikes me as yet another reason the correct spelling wasn't widely adopted.
- Regards, DIV 22.214.171.124 07:04, 30 January 2007 (UTC)
- http://scholar.google.com/ has 0 hits on "Sierpiński number" and 10 on "Sierpinski number". http://www.altavista.com/ has 0 on "Sierpiński number" and 570 on "Sierpinski number". Both have hits on "Sierpiński" alone so they recognize ń. If there isn't a single search engine hit saying "Sierpiński number" then I don't think the article main text should use it either (and it doesn't although you say so). It sounds like WP:OR violation to say the "correct" name is something other than EVERYBODY uses. Redirect rules are less strict so it's okay to keep a redirect on Sierpiński number. PrimeHunter 15:39, 30 January 2007 (UTC)
Importance and applications
Why are Sierpinski numbers important? What are they used for?
- I explained it in . Is that OK? PrimeHunter (talk) 15:00, 12 February 2008 (UTC)
Funky question mark
Why is there a funky question mark in the definition of the problem? In the section "The Sierpinski problem" It looks ridiculous and it isn't in the style of wikipedia. I think that silly box should be removed entirely. 126.96.36.199 (talk) 19:00, 15 October 2010 (UTC)
- It's made by Template:Unsolved. The template was kept at Wikipedia:Templates for discussion/Log/2010 September 29#Template:Unsolved. PrimeHunter (talk) 19:29, 15 October 2010 (UTC)
The divination of Sierpinski number
The user "Prime Hunter" said: "We define Sierpinski number as odd k; an even k corresponds to an odd k with a larger n: (2^m*j)*2^n+1 = j*2^(m+n)+1" but I think that it is not always. For example, if there are only five Fermat primes: 3, 5, 17, 257, 65537, than 131072 and larger powers of 2 are Sierpinski numbers, is it right? — Preceding unsigned comment added by 188.8.131.52 (talk)
- I'm not sure what you are asking. It's right that we define a Sierpiński number as odd k. It's in the opening sentence. You are quoting an edit summary  I gave as reason for not including even k in a table of smallest n for which k×2n+1 is prime. I don't see reason to include even k when the article is about Sierpiński numbers. Every prime of form k×2n+1 with even k corresponds to a prime of the same form with a larger odd k. But if even numbers were allowed as Sierpiński numbers then there would probably be some even Sierpiński numbers which don't correspond to an odd Sierpiński number. Specifically, it would occur in any case where there is a finite non-zero number of primes of form k×2n+1 for an odd k. Those k would not be Sierpiński numbers, but if the largest prime is k×2m+1 then k×2n would be a Sierpiński number for all n > m. I don't know of any reliable source which has considered this definition so it shouldn't be in the article. PrimeHunter (talk) 11:16, 22 May 2014 (UTC)
I was asking for that k=131072, if there are only 5 Fermat primes, 3, 5, 17, 257 and 65537, than 131072 is a Sierpinski number. — Preceding unsigned comment added by 184.108.40.206 (talk) 05:20, 4 June 2014 (UTC)
- If we change the definition to allow even Sierpinski numbers then yes, if 65537 is the largest Fermat prime then 131072 and all larger powers of two would be Sierpinski numbers. But we shouldn't change the definition to something not found in any reliable sources. PrimeHunter (talk) 23:45, 4 June 2014 (UTC)
Can k be any integers?
k*2^n+1 or 2^n+k
78557*2^n+1 must divide by 3, 5, 7, 13, 19, or 37, and 2^n+78557 as well, but all odd numbers under 78557 expect 40291 have a prime in a form 2^n+k, so 78557 is the smallest Sierpinski number. — Preceding unsigned comment added by 220.127.116.11 (talk) 14:18, 30 June 2014 (UTC)