Talk:Sierpiński number

Selfridge's conjecture
Selfridge's conjecture as [defined at MathWorld] is not the same as the definition here. --85.194.2.201 15:51, 20 December 2005 (UTC)

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 &equiv; 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&lt;78557, but the last few proved difficult, so the holdouts are being attacked by simple brute force. 71.41.210.146 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 128.250.204.118 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 128.250.204.118 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 Naming conventions (common names). I didn't find a single source saying "Sierpiński number" and lots saying "Sierpinski number". See also Naming conventions (use English) (the term "Sierpinski number" may be an English invention and not a native name in Wacław Sierpiński's language) and 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 128.250.204.118 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?

Solved k
Can anyone who knows the subject clarify it? It is not clear what is meant by Solved k.Hakeem.gadi (talk) 09:30, 12 February 2008 (UTC)
 * I explained it in . Is that OK? PrimeHunter (talk) 15:00, 12 February 2008 (UTC)
 * Yup, that's alright. Thanks.Hakeem.gadi (talk) 07:01, 18 February 2008 (UTC)

Requested move

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the move request was: no consensus. harej 21:16, 6 June 2010 (UTC)

Sierpinski number → — Diacritics should not be left out in titles. I cannot move the page because Sierpiński number redirects to Sierpinski number. --Number Googol (talk) (my edits) 20:26, 24 May 2010 (UTC)
 * Oppose. Diacritics should be left out when, as here, English generally omits them. Septentrionalis PMAnderson 22:29, 24 May 2010 (UTC)


 * Oppose per Septentrionalis, Article titles, Naming conventions (use English), and my arguments above at . Nearly all sources say "Sierpinski number". Wikipedia shouldn't change something sources agree about. Many things named after people are spelled differently in English than the person in their own language and alphabet. This is not a biography but an English article about a mathematical term. PrimeHunter (talk) 23:59, 24 May 2010 (UTC)
 * Google Scholar, which you cite in your arguments above, doesn't distinguish between Sierpiński number and Sierpinski number. Searching for "Sierpiński number" gives 27 hits, as does searching for "Sierpinski number". --Atemperman (talk) 00:08, 25 May 2010 (UTC)
 * They distinguished in 2007 during the old discussion. It appears "Sierpiński number" is increasing in frequency but still a minority of English sources. http://www.altavista.com gives me 41 hits on "Sierpiński number" and 1560 on "Sierpinski number". PrimeHunter (talk) 00:53, 25 May 2010 (UTC)
 * Especially for something technical like Sierpiński numbers, what people type on the internet should be given far less weight than what appears in authoritative, reliable works of reference or edited publications. These include Britannica and Mathworld, both of which use "Sierpiński number".  On Google Scholar, I looked in the first page of hits, of which four are papers in reviewed, edited journals.  Three out of the four of these sources use "Sierpiński number"; the one that isn't is from 1983; the other three are more recent.  Two of the sources are edited but not reviewed works; there the split is 1-1.  The remaining four sources have three out of four using "Sierpiński number" rather than "Sierpinski number".  So in sum, it's 7-3 in favor of "Sierpiński number", with the split among the most authoritative works 3-1 in favor.
 * It's also worth noting that you have to actually click on the links and investigate how the term appears rather than relying on Google's preview -- most of the time, the preview renders as "Sierpinski number" what is actually written as "Sierpiński number" in the source. This makes me doubt whether the Altavista numbers can be trusted. --Atemperman (talk) 13:09, 25 May 2010 (UTC)
 * Support. Of the three references, two use Sierpiński rather than Sierpinski; one of these is an authoritative reference work. The other reference is post on a message board.  Moreover, Britannica writes Sierpiński as well.--Atemperman (talk) 00:01, 25 May 2010 (UTC)
 * Either way, the titles of all Wikipedia articles related to Wacław Sierpiński should be consistently spelled using either Sierpiński or Sierpinski. Currently, on Wikipedia, Sierpinski triangle and Sierpinski carpet use "Sierpinski", while Sierpiński space, Sierpiński's constant, Sierpiński curve, and Sierpiński arrowhead curve use "Sierpiński". --Number Googol (talk) (my edits) 04:08, 25 May 2010 (UTC)
 * Usage should prevail. It probably should be Sierpinski for all of these; but different fields may differ - and then there's the question of which editors we have. Septentrionalis PMAnderson 20:22, 25 May 2010 (UTC)
 * Septentrionalis has made assertions without any evidence. I've provided evidence that Sierpiński is predominant in authoritative sources and have rebutted PrimeHunter's arguments in favor of Sierpinski.  I hope whoever the admin is who makes the decision takes note of this.  --Atemperman (talk) 02:16, 28 May 2010 (UTC)


 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

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. 173.13.109.1 (talk) 19:00, 15 October 2010 (UTC)


 * It's made by Template:Unsolved. The template was kept at Templates for discussion/Log/2010 September 29. 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 140.113.136.218 (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 [//en.wikipedia.org/w/index.php?title=Sierpinski_number&diff=604146283&oldid=604105164] 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 140.113.136.220 (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?
We can define Sierpinski numbers on all the integers, such as -509203, a negative integer. — Preceding unsigned comment added by 115.80.80.42 (talk) 06:25, 27 June 2014 (UTC)

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 180.204.20.28 (talk) 14:18, 30 June 2014 (UTC)

Strange sentences
I presume that there are syntax errors in the following sentences :
 * "However, some values of ns are large, for example, the smallest solution for that k = 2131, 40291, and 41693, the least n are 4583176, 9092392, and 5146295. (However, the least n such that k2n + 1 is only 44, 8, and 33. Interestingly, the least n which 2n + 10223 is prime is only 19.)"

There are other strange sentences. Marvoir (talk) 17:22, 15 January 2016 (UTC)

Sierpinski Problems
I have split the section in 3, to mention and detail the other two SPs, I updated the lists (some from 5 years ago). A native speaker is welcomed to correct my English (in spite of the fact that I took part of the text from protsearch.com, :P) — Preceding unsigned comment added by LaurV (talk • contribs) 04:53, 23 May 2017 (UTC)

Some genius eliminated the "Prime" problem. In this respect, the "Extended" problem that I added some time ago (see the paragraph above) is now hanging with "Suppose that both preceding Sierpiński problems had finally been solved", but which both? there is only one described... LaurV (talk) 04:57, 23 November 2018 (UTC)
 * I put it back... somebody may clean it, but please do not remove it. LaurV (talk) 06:54, 23 November 2018 (UTC)

Specification
I think it must be said there is a sequence $$ k \times 2^n + 1 $$ involved in the definition of a Sierpinski number.--109.166.132.148 (talk) 18:38, 16 April 2020 (UTC)
 * The definition does not require a sequence and none of the sources I examined mention a sequence. The current definition seems clear to me. It's about all numbers of form $$ k \times 2^n + 1 $$. There is no need to consider the set of such numbers as a sequence with increasing n values.  PrimeHunter (talk) 18:49, 16 April 2020 (UTC)
 * Why not? These numbers of this form $$s_n = k \times 2^n + 1 $$  naturally constitute a sequence or equivalently, an ordered set because the order of numerical elements in this set is NOT an arbitrary/random one.--109.166.132.148 (talk) 19:07, 16 April 2020 (UTC)
 * Yes, it has a natural order. But this is irrelevant to anything interesting about it.  So why add pointless complication? --JBL (talk) 19:09, 16 April 2020 (UTC)
 * I do not think it would be a pointless complication. A certain member of this ordered set can be specified of a given k, like the 3rd Sierpinski number, the 4th, the 5th, the jth Sierpinski number, etc.. Any elements with an number of order (index or rank) form a sequence whose individual terms may have a certain property assigned to it, such as divisible to a certain m number, can be involved in modulo congruences, etc.--109.166.132.148 (talk) 19:20, 16 April 2020 (UTC)
 * Also individual terms of the sequence can be counterexamples in some reasonings.--109.166.132.148 (talk) 19:23, 16 April 2020 (UTC)
 * This context re counterexamples leads to the question of existance of at least of a certain k for which the terms of the sequence may be prime numbers.--109.166.132.148 (talk) 19:28, 16 April 2020 (UTC)
 * This is utterly unconvincing. --JBL (talk) 21:31, 16 April 2020 (UTC)
 * What exactly is unconvincing?--109.166.132.148 (talk) 23:32, 16 April 2020 (UTC)
 * Everything you've written. None of it is related to the all-important question how best to present the information about Sierpinski numbers that exists in reliable sources in an encyclopedic way? --JBL (talk) 23:50, 16 April 2020 (UTC)
 * What exactly would be un-encyclopedic in mentioning the word sequence in relation to Sierpinski numbers? Above you claim that adding the word sequence would be a pointless complication, which is clearly an exaggeration. I don't quite understand the claimed/supposed status of pointless complication, clearly is not the case, what complication generated by just using a word? I think that this claimed status is far from being convincing, to use your wording.--109.166.132.148 (talk) 01:23, 17 April 2020 (UTC)
 * I do not see any purpose in continuing this discussion; your edit was an obvious disimprovement for the reasons I’ve mentioned, please don’t repeat it. —JBL (talk) 01:51, 17 April 2020 (UTC)
 * Obvious disimprovement? This seems to be a joke! When you say/claim that something is unencyclopedic, you should explain your claim! I do not understand your reaction, no need to become hostile.--109.166.132.148 (talk) 09:51, 17 April 2020 (UTC)
 * Wikipedia is based on reliable sources and mathematicians prefer definitions without unnecessary parts. I have now examined ten definitions of Sierpinski number. The all say almost exactly the same as us. None of them use the word sequence. They have no reason to, and neither do we. PrimeHunter (talk) 03:26, 17 April 2020 (UTC)
 * You could check again an additional source. There is a source right in the article which mentions the word sequence. The use of the word sequence is just explanatory/descriptive for a clearer understanding, not really part of a strict definition, just something that follows as a (noticeable) property from the number of order, thus it can be added by WP:CALC.--109.166.132.148 (talk) 09:51, 17 April 2020 (UTC)
 * (What is almost exactly said by those ten sources? What is the difference from what is in the article? Could those sources def be listed here?)--109.166.132.148 (talk) 10:35, 17 April 2020 (UTC)
 * Please, stop repeating yourself, this is not WP:CALC it is WP:IDHT. Your eccentric views are not compelling at all, and you do not seem to understand what is the purpose of the lead sentence. --JBL (talk) 10:44, 17 April 2020 (UTC)
 * What repetitions? It is not WP:IDHT. I have underlined the explanatory nature of the discussed word in re to the strict definition (without unnecessary parts) of the Sierpinski numbers (this aspect has not been mentioned before, therefore NO repetition). Also WP:CALC has not been mentioned before 09:51, 17 April 2020. How does the purpose of the lead sentence get negatively impacted by the use of a certain word? Please clarify!--109.166.132.148 (talk) 12:23, 17 April 2020 (UTC)
 * There is also an open question from the 21:14 10 September 2019 edit which hasn't been adressed in re to the proposed word, namely (what is "the sequence" supposed to refer to?) .--109.166.132.148 (talk) 12:23, 17 April 2020 (UTC)
 * The lead sentence would get longer and more complicated, introduce a term some readers don't know or are unsure about the technical meaning, introduce a notation $$s_n$$ not used in the rest of the article or sources, and give a misleading impression that Sierpinski numbers are normally defined in terms of a sequence. You are not going to convince us there is any good reason for this. Just Google Sierpinski number to find many definitions. They may say "not prime" instead of composite and have a different sentence structure but that's about it as far as differences go. They even use the same variable names. PrimeHunter (talk) 12:55, 17 April 2020 (UTC)
 * This update with clear explanations from 12:55, 17 April 2020 is much more convincing or has a higher argumentative power/value to the choice NOT to use the proposed word than the initial dry labels (pointless complication, utterly unconvincing, uncyclopedic, stop repeating yourself, eccentric views, etc) which seemed unnecessarily hostile.--109.166.132.148 (talk) 16:57, 17 April 2020 (UTC)

Requested move 31 August 2020

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion. 

The result of the move request was: Page moved. (non-admin closure)  Vpab15 (talk) 09:46, 8 September 2020 (UTC)

Sierpinski number → Sierpiński number – 10 years ago this might be a 'no consensus', but in the decade since, the usage of diacritics has become the norm. The article also uses the diacritic ń everywhere, just not in the very name. Let's standardize this and move on (pun intended). Also: Wacław Sierpiński, Sierpiński triangle, Sierpiński curve, Sierpiński's constant, Sierpiński set, Sierpiński space, and Sierpiński carpet. Piotr Konieczny aka Prokonsul Piotrus&#124; reply here 07:37, 31 August 2020 (UTC)
 * Yes sure. —JBL (talk) 11:44, 31 August 2020 (UTC)


 * Support per nom.--Ortizesp (talk) 13:48, 31 August 2020 (UTC)
 * Yea, why not? Bubba73 You talkin' to me? 20:43, 31 August 2020 (UTC)
 * Oppose per WP:COMMONNAME The use of the diacritic is not the most common English spelling according to the Google Ngrams. Rreagan007 (talk) 22:47, 31 August 2020 (UTC)
 * Historical use is not very helpful as until relatively recently, most printing presses had technical trouble with non-standard characters. But this is the 21st century. --Piotr Konieczny aka Prokonsul Piotrus&#124; reply here 03:11, 1 September 2020 (UTC)
 * Support per WP:CONSISTENCY. I wouldn't trust the Ngrams result as it claims that the ngram "Sierpiński number" does not exist at all, which is patently false. -- King of ♥ ♦ ♣ ♠ 01:44, 1 September 2020 (UTC)
 * It finds Sierpinski and Sierpiński. Perhaps "Sierpiński number" is just too uncommon in English to register. Rreagan007 (talk) 02:50, 1 September 2020 (UTC)
 * I think it is more likely ngram results are not coded well enough to distinguish between texts with diacritics and without. I've seen this with some OCR results, where some, particularly older, OCR software will either render ń as n or as something else like (ñ, ǹ, etc.). --Piotr Konieczny aka Prokonsul Piotrus&#124; reply here 03:14, 1 September 2020 (UTC)


 * Support WP:CONSISTENCY with the entire en.wp article corpus, and with current Serbian name RMs. In ictu oculi (talk) 12:06, 1 September 2020 (UTC)
 * Oppose per WP:COMMONAME. -- GoodDay (talk) 14:28, 1 September 2020 (UTC)
 * Support I thought we got over all the hand wringing over diacritics a decade ago? This should be non-controversial. Wolfram uses "Sierpiński", so does almost any non-outdated sources that uses anything more advanced than bare text. It's the correct way to do it, especially for an encyclopedia. Also WP:CONSISTENCY.  Volunteer Marek   15:15, 1 September 2020 (UTC)
 * Support for consistency with the other articles. kennethaw88 • talk 23:24, 3 September 2020 (UTC)


 * The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Powers of Sierpinski numbers
According to an explanation from Prof. Michael Filaseta posted on the SeqFan list ("Re: is it true", Jul 12 2022), if k is Sierpinski with a known "covering set", then so is k^m for all m relatively prime to the LCM of the covering set. I think this is worth mentioning. But is this a known result? Has anyone a published reference or similar? &mdash; MFH:Talk 17:35, 23 July 2022 (UTC)

My related conjecture
Like Bunyakovsky conjecture for polynomial sequences, I have a conjecture about exponential sequences:

If a ≥ 1 is integer, b ≥ 2 is integer, c ≠ 0 is integer, $$gcd(a,c)=1$$, $$gcd(b,c)=1$$, and $$\frac{a \cdot b^n+c}{gcd(a+c,b-1)}$$ cannot be proven to only contain composites or only contain finitely many primes by covering congruence (such as $$122 \cdot 13^n+3$$, which is always divisible by 5, 7, or 17), algebraic factorization (such as $$8 \cdot 27^n+1$$, which has sum-of-two-cubes factorization), or combine of them (such as $$25 \cdot 12^n -1$$, which is divisible by 13 if n is odd and has difference-of-two-squares factorization is n is even), then there are infinitely many integers n ≥ 1 such that $$\frac{a \cdot b^n+c}{gcd(a+c,b-1)}$$ is prime. 36.233.231.45 (talk) 04:26, 19 June 2023 (UTC)


 * Wikipedia is an encyclopedia; it is for reporting & recording what is found in high-quality published sources. Wikipedia talk-pages are for proposing and discussing improvements to Wikipedia articles, within that framework.  Wikipedia is not a venue for promoting, publishing, or discussing original research, including new conjectures.  --JBL (talk) 17:18, 19 June 2023 (UTC)