![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
The two "citation needed" things are definitely true but the only references I have are to out-of-print books.
More on "applications" is called for.
I dunno why the software generates two different fonts for the formulas. Somebody fix it, would you? :) LDH ( talk) 03:03, 25 November 2008 (UTC)
It could be fixed by having all rendered by LaTeX (or all as HTML). To do the former just add \, at the end of each formula (a spacer which can not be rendered in HTML). —Preceding unsigned comment added by 208.63.161.184 ( talk) 15:04, 9 May 2009 (UTC)
I just added a request for clean-up because of the monstrous paragraph at the end of the Properties section is a rambling mess. Additionally, it contains a reference to a very recent record-height polynomial. Computing such large polynomials requires dozens of hours of computation time and many gigabytes of memory, so this is probably original research. 142.58.12.53 ( talk) 20:31, 24 January 2011 (UTC)
Yes, it IS original research, and I added the source now in two weblinks. Maybe there is a mistake in formatizing, because the URLs begin Www.... ( talk) 22:19, 25 January 2011 (UTC)
I'm concerned that this assertion [in `Fundamental Tools']
> The Möbius inversion formula yields the equivalent formulation: > > \prod_{d\mid n}(x^d-1)^{\mu(n/d)} = \Phi_n(x) >
is QUITE FALSE?! It seems to imply, for example, that \Phi_n(1) = 0 for any n, whereas the first assertion on the page
> If n is a prime number then > > ~\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1} x^i.
implies that \Phi_p(1) = p if n = p is prime.
The mistake seems to originate in a misapplication of the M\"obius inversion formula, which would require that
> \Phi_{mn}(x) = \Phi_m(x)\Phi_n(x)
for m,n coprime; in other words, that \Phi_n is a `multiplicative' arithmetic function. This fails, for example, if m = 2 and n is an odd prime.
This error seems to be recur in the article on M\"obius inversion, where the requirement that the function be multiplicative is not mentioned. This property is sometimes taken as part of the definition of an `arithmetic function', which may be where this error crept in.
++++++++++++++++++++++++
I hope I am mistaken. I am inexperienced in the ways of Wikipedia, and suspect this entry requires the attention of an expert in number theory = which I am not. I hope the readers, if any, of this message will bring it to the attention of some such person.
Thanks for your attention! — Preceding unsigned comment added by Drwonmug ( talk • contribs) 18:01, 28 January 2013 (UTC)
The section "Prime Cyclotomic numbers", introduced in April 2014 by an IP user was essentially original research, consisting essentially in reproducing and commenting some OEIS tables. As these tables were relevant to the subject, I did not removed the section, and I have only tagged it as unreferenced. A reason for not removing the section was that there were no other section devoted to the study of the values of cyclotomic polynomials.
Recently, when trying to wikify Unique prime (an article probably introduced by the same IP user), I found interesting properties of the values of the cyclotomic polynomials, which were stated (partially incorrectly) without explanation nor reference. As I have no access to the references of Unique prime, I have provided a proof of these properties. I ignore if they were published elsewhere, but, as the proof is rather elementary, this is highly probable. As these properties are relevant for a section of the values of cyclotomic polynomials, I have introduced them as a new section § Values, which contains also links to the OEIS tables of the preceding section.
References for these results would be welcome. D.Lazard ( talk) 15:47, 21 October 2016 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on Cyclotomic polynomial. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
RfC before doing mass systematic removals. This message is updated dynamically through the template {{
source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 02:04, 16 August 2017 (UTC)
The example sections gives two formulas, one for when n is prime, and the other for n is 2p, where p is an odd prime. But it doesn't give a formula for when neither condition applies. Could somebody add this? MiguelMunoz ( talk) 06:46, 17 September 2023 (UTC)
![]() | This article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
The two "citation needed" things are definitely true but the only references I have are to out-of-print books.
More on "applications" is called for.
I dunno why the software generates two different fonts for the formulas. Somebody fix it, would you? :) LDH ( talk) 03:03, 25 November 2008 (UTC)
It could be fixed by having all rendered by LaTeX (or all as HTML). To do the former just add \, at the end of each formula (a spacer which can not be rendered in HTML). —Preceding unsigned comment added by 208.63.161.184 ( talk) 15:04, 9 May 2009 (UTC)
I just added a request for clean-up because of the monstrous paragraph at the end of the Properties section is a rambling mess. Additionally, it contains a reference to a very recent record-height polynomial. Computing such large polynomials requires dozens of hours of computation time and many gigabytes of memory, so this is probably original research. 142.58.12.53 ( talk) 20:31, 24 January 2011 (UTC)
Yes, it IS original research, and I added the source now in two weblinks. Maybe there is a mistake in formatizing, because the URLs begin Www.... ( talk) 22:19, 25 January 2011 (UTC)
I'm concerned that this assertion [in `Fundamental Tools']
> The Möbius inversion formula yields the equivalent formulation: > > \prod_{d\mid n}(x^d-1)^{\mu(n/d)} = \Phi_n(x) >
is QUITE FALSE?! It seems to imply, for example, that \Phi_n(1) = 0 for any n, whereas the first assertion on the page
> If n is a prime number then > > ~\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1} x^i.
implies that \Phi_p(1) = p if n = p is prime.
The mistake seems to originate in a misapplication of the M\"obius inversion formula, which would require that
> \Phi_{mn}(x) = \Phi_m(x)\Phi_n(x)
for m,n coprime; in other words, that \Phi_n is a `multiplicative' arithmetic function. This fails, for example, if m = 2 and n is an odd prime.
This error seems to be recur in the article on M\"obius inversion, where the requirement that the function be multiplicative is not mentioned. This property is sometimes taken as part of the definition of an `arithmetic function', which may be where this error crept in.
++++++++++++++++++++++++
I hope I am mistaken. I am inexperienced in the ways of Wikipedia, and suspect this entry requires the attention of an expert in number theory = which I am not. I hope the readers, if any, of this message will bring it to the attention of some such person.
Thanks for your attention! — Preceding unsigned comment added by Drwonmug ( talk • contribs) 18:01, 28 January 2013 (UTC)
The section "Prime Cyclotomic numbers", introduced in April 2014 by an IP user was essentially original research, consisting essentially in reproducing and commenting some OEIS tables. As these tables were relevant to the subject, I did not removed the section, and I have only tagged it as unreferenced. A reason for not removing the section was that there were no other section devoted to the study of the values of cyclotomic polynomials.
Recently, when trying to wikify Unique prime (an article probably introduced by the same IP user), I found interesting properties of the values of the cyclotomic polynomials, which were stated (partially incorrectly) without explanation nor reference. As I have no access to the references of Unique prime, I have provided a proof of these properties. I ignore if they were published elsewhere, but, as the proof is rather elementary, this is highly probable. As these properties are relevant for a section of the values of cyclotomic polynomials, I have introduced them as a new section § Values, which contains also links to the OEIS tables of the preceding section.
References for these results would be welcome. D.Lazard ( talk) 15:47, 21 October 2016 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on Cyclotomic polynomial. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
RfC before doing mass systematic removals. This message is updated dynamically through the template {{
source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 02:04, 16 August 2017 (UTC)
The example sections gives two formulas, one for when n is prime, and the other for n is 2p, where p is an odd prime. But it doesn't give a formula for when neither condition applies. Could somebody add this? MiguelMunoz ( talk) 06:46, 17 September 2023 (UTC)