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Archiving 19 discussion(s) from Talk:Euler's totient function) (bot |
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::::: No doubt: lowercase \phi. But indeed, especially on computers, I think the "straight" version (ϕ) is more frequent than the handwritten one, which in turn is preferred for angles (I think).— [[User:MFH|MFH]]:[[User talk:MFH|Talk]] 13:16, 3 June 2008 (UTC)
:An italic html phi looks more like the tex angled one: ''φ''. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 15:14, 16 June 2013 (UTC)
==Untitled==
The use of eight (8) for n (the argument to the totient function) in the opening paragraph might be misleading to some. All of it's coprimes are also prime. The use of nine (9) as the argument might be less misleading since it includes two composite numbers in it's set of coprimes. The number nine is also a power of a prime (just like eight) and thus allows simple computation of the totient. -[[User:Waylonflinn|Waylonflinn]] 22:38, 28 December 2006 (UTC)
How to pronounce totient?—[[User:Tokek|Tokek]] ([[User talk:Tokek|talk]]) 08:15, 23 June 2005 (UTC)
:Totient rhymes with division's quotient. —[[User:Optikos|optikos]] ([[User talk:Optikos|talk]]) 03:48, 24 June 2008 (UTC)
I think it is "toe shent". BTW, I prefer "the cototient is defined as" because it is a definition, not something that happens to be true. [[User:Bubba73|Bubba73]] 15:22, 23 June 2005 (UTC)
: Both "is defined as" and "is" are appropriate, however you actually typed "defined as is" instead of "is defined as" so it required fixing. The statement can be taken as a definition for cototient either way.—[[User:Tokek|Tokek]] ([[User talk:Tokek|talk]]) 17:24, 23 June 2005 (UTC)
:: That was a typo. "is defined as" makes it clear that it is a definition. If they're both appropriate, why not use the one that is more appropriate (is defined as)? I wrote that line originally without the "defined" part. Then I realized that it would improve the article to have "defined" in there, so I changed it, but I accidently stuck it in at the wrong place. [[User:Bubba73|Bubba73]] 18:01, 23 June 2005 (UTC)
::: Noted. —[[User:Tokek|Tokek]] ([[User talk:Tokek|talk]]) 19:06, 23 June 2005 (UTC)
Doesn't totient(n) always equal a positive integer? [[User:Railgun|Railgun]] 16:59, 26 June 2006 (UTC)
What does the phrase "randomly large n" mean? Should that be "general n"? [[User:62.8.160.190|62.8.160.190]] 05:17, 26 July 2006 (UTC)
== Notation - needs editing ==
In the section titled History, Terminology and notation, the notation is NOT explained.
That is really really lame. A link is provided to Arithmetic Functions where the notation is only partially explained. This link is not in the notation section!! More massive lameness. If you are going to write it, explain it.
Gamma d|n and Sigma d|n are explained, but a|b is NOT explained nor is phi(n)|phi(m).
In the linked article the explanation of the notation linked to is just two lines. WTF wasn't it included here?? I would cut and paste it, but since I haven't the faintest idea what a|b means, perhaps one of the illuminati can condescend?
<math>\sum_{d|n} f(d)\;</math> and <math>\prod_{d|n} f(d)\;</math> mean that the sum or product is over all positive divisors of ''n'', including 1 and ''n''. E.g., if ''n'' = 12,
:<math>\prod_{d|12} f(d) = f(1)f(2) f(3) f(4) f(6) f(12).\ </math>
p|n also needs explanation as an index = all prime divisors of n. And possibly the Fourier Transform notation...
Thanks?[[Special:Contributions/72.172.11.228|72.172.11.228]] ([[User talk:72.172.11.228|talk]]) 17:20, 28 May 2013 (UTC)
:In addition, if the notation is as described on the Arithmetic Functions page as shown in the product above (i.e. it includes 1 and ''n'') then shouldn't it follow that:
::<math>\prod_{p|n} (1-\frac 1 p) = (1- \frac 1 1) \times \cdots \times (1 - \frac 1 n) = 0.\ </math>
:That is pretty clearly incorrect, but it is what a literal reading of the notation description would lead one to believe.<small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:128.138.65.99|128.138.65.99]] ([[User talk:128.138.65.99|talk]] • [[Special:Contributions/128.138.65.99|contribs]]) 15:10, 30 May 2014</span></small><!-- Template:Unsigned -->
::The article says "where the product is over the distinct prime numbers dividing n". As 1 is not a prime number, <math>(1-\frac 1 1)</math> cannot appear in the product, and the product is never zero. Similarly, the factor <math>(1-\frac 1 n)</math>, may appear only if ''n'' is prime. In this case, it is the only factor.
[[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:56, 30 May 2014 (UTC)
== Calligraphic O ==
I suppressed the calligraphic O for the Landau symbol. This font is simply never used in number theory in the literature (in computer science I don't know). I've seen it only in some wikipedia pages… [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 22:36, 7 July 2014 (UTC)
:I've seen it used in computer science but it's not universal there. Anyway this article is number theory not CS. So I agree with this decision. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 23:29, 7 July 2014 (UTC)
== Definition of φ(0) ==
The edit summary for [https://en.wikipedia.org/w/index.php?title=Euler%27s_totient_function&diff=prev&oldid=630824632 this reversion] says: "The function is not defined for ''n''=0." That statement should be in the article and sourced. See [http://books.google.com/books?id=3hTeH5VUheAC Hardy & Wright]: 5.5 ''Euler's function φ(m)''.
The edit summary for [https://en.wikipedia.org/w/index.php?title=Euler%27s_totient_function&diff=next&oldid=630824632 this subsequent reversion] says: "Mathematica defines EularPhi(0) as 0". That statement should also be in the article and an explanation given. See the [[Mathematica]] documentation for [http://reference.wolfram.com/language/ref/EulerPhi.html EulerPhi] and this sentence from [[MathWorld]]:
*"By convention, phi(0)=1, although Mathematica defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command."
:({{MathWorld|id=TotientFunction|title=Totient Function}})
--[[Special:Contributions/50.53.60.172|50.53.60.172]] ([[User talk:50.53.60.172|talk]]) 07:51, 24 October 2014 (UTC)
:I find the assertion "By convention, phi(0)=1" very suspect; I am a number theorist and I have never heard of such a convention before. In general I think the informations from "Mathworld" should be taken with care, especially those pertaining to definitions and history, and verified elsewhere before they are included in Wikipedia: "Mathworld" is also an online encyclopedia, but with essentially one single contributor, who is not originally a mathematician (nor a historian).
:Regarding the fact that Mathematica provides the value 0 for phi(0) (or any value): it appears to me as a bug, that should be removed as soon as possible. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 08:45, 24 October 2014 (UTC)
::I agree with Sapphorain. Moreover, giving a value for phi(0) in the table would implies changing accordingly the first paragraph of the lead and making it unsourced, as the sources 1 and 2 define the totient function only for ''positive'' integers. As these sources are highly reliable on this subject, much more than Mathworld and Matematica, such a change would break two Wikipedia policies: [[WP:Verifiability]] and [[WP:Neutral point of view]]. — [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:28, 24 October 2014 (UTC)
::#The article should [[WP:OBVIOUS|explicitly state]] that φ(0) is undefined by Hardy & Wright and any other number theory [[Wikipedia:Reliable sources|sources]] you care to cite. The table should conform to that.
::#A [[Bug (software)|bug]] is a mistake. The Mathematica convention is not a bug. The Mathematica documentation for [http://reference.wolfram.com/language/ref/EulerPhi.html EulerPhi] says EulerPhi[0] returns 0. (look under "Possible Issues"). The article should acknowledge that fact — this is an encyclopedia, not a number theory textbook.
::#[http://mathworld.wolfram.com/TotientFunction.html The MathWorld statement] that "By convention, phi(0)=1" is not explicitly sourced, despite a huge reference section, so MathWorld is unreliable on that point.
:::--[[Special:Contributions/50.53.60.172|50.53.60.172]] ([[User talk:50.53.60.172|talk]]) 13:20, 24 October 2014 (UTC)
::::Clarification: Hardy & Wright <u>do not explicitly state</u> that φ(0) is undefined. In their definition of φ({{mvar|m}}), the [[Domain (mathematics)|___domain]] of φ is implied by their inequality:
::::*<math>0 < n \le m</math> (§ 5.5, p. 52)
::::--[[Special:Contributions/50.53.60.172|50.53.60.172]] ([[User talk:50.53.60.172|talk]]) 13:58, 24 October 2014 (UTC)
:::Here is a simple solution for the table: Add a note above the table that [[WP:OBVIOUS|explicitly states]] that φ(0) is undefined by various [[WP:CITE|cited]] sources. --[[Special:Contributions/50.53.60.172|50.53.60.172]] ([[User talk:50.53.60.172|talk]]) 14:18, 24 October 2014 (UTC)
::::The issue here is not how phi(0) should be defined, but whether is should be defined at all. For a number of very good reasons, related to the underlying structures of the sets of arithmetical functions and of the multiplicative arithmetical functions, to which the phi function belongs, it should definitely not be defined. Indeed these are groups under the [[Dirichlet convolution]]; but, as every positive integer is a divisor of 0, the Dirichlet convolution f*g(n) is undefined if n=0. Thus, for f an arithmetical function, f(0) is never defined in good textbooks, in which these underlying structures are studied: the set of arguments for f is the set of of positive integers. So there is no point in providing sources in which phi(0) is not defined: no serious source will define it.
::::Maybe the value for phi(0) provided by Mathematica is not, strictly speaking, a bug, as it appears to be intentional. But being intentional doesn't change the fact that it is a mistake. Besides, we don't even know ''who'' decided to provide this strange output, and ''why''. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 16:40, 24 October 2014 (UTC)
:::::{{tq|"The issue here is not how phi(0) should be defined, but whether is should be defined at all."}}
:::::You are wasting your time explaining why or why not φ(0) should be defined. WP requires [[WP:V|verifiable]] and [[WP:RS|reliable]] sources, so it would be more constructive if you provided some sources making your point.
:::::And the article is [[WP:POV|biased]], because it does not say what [[Mathematica|a highly-respectable software product]] does. The MathWorld article offers an explanation for why EulerPhi[0] returns 0:
:::::*"... Mathematica defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command." [http://mathworld.wolfram.com/TotientFunction.html (link)]
:::::Do you know what that means?
:::::For your convenience, here are links to the [[Mathematica]] documentation for [http://reference.wolfram.com/language/ref/EulerPhi.html EulerPhi] and [http://reference.wolfram.com/language/ref/FactorInteger.html FactorInteger].
:::::--[[Special:Contributions/50.53.60.172|50.53.60.172]] ([[User talk:50.53.60.172|talk]]) 17:39, 24 October 2014 (UTC)
:::::: Yes, indeed, I think I have been wasting my time. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 21:07, 24 October 2014 (UTC)
:{{tq|'..."Mathematica defines EularPhi(0) as 0". That statement should also be in the article and an explanation given.'}}
:EulerPhi also accepts negative integers. For example, [[WolframAlpha]] shows that [https://www.wolframalpha.com/input/?i=EulerPhi%5B-60%5D EulerPhi[-60<nowiki>]</nowiki>] is 16.
:--[[Special:Contributions/50.53.60.172|50.53.60.172]] ([[User talk:50.53.60.172|talk]]) 03:20, 25 October 2014 (UTC)
::This is becoming quite ridiculous. How about complex values of the argument? If you wish to mention an extension of the Euler function defined only by "a highly respectable software product" and considered nowhere else, I think you'll have to do it in a footnote. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 08:01, 25 October 2014 (UTC)
:::{{ec}} This simply confirm that Mathematica is not a reliable source. No need of wasting our time for that. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 08:07, 25 October 2014 (UTC)
::::{{ut|Sapphorain}}: {{tq|"... an extension of the Euler function ..."}}
::::That's an excellent way to describe the ___domain of the <u>software function</u> EulerPhi. Haven't any mathematicians discussed the [[Domain (mathematics)|___domain]] of the <u>mathematical function</u> φ? A footnote about EulerPhi would be fine, if the article clearly stated that <u>mathematicians</u> do not define {{math|φ(''n'')}} for {{math|''n'' < 1}} and explained why. (It has something to do with the solution to the equation {{math|[[Greatest common divisor|gcd]](''n'', ''k'') {{=}} 1}} for {{math|''n'' < 1}}.)
::::{{ut|D.Lazard}}: {{tq|"... Mathematica is not a reliable source."}}
::::The [[Mathematica]] documentation is a [[wp:rs|reliable source]] for what the <u>software function</u> [http://reference.wolfram.com/language/ref/EulerPhi.html EulerPhi] does. Indeed, the documentation says:
::::*"{{math|φ(-''n'')}} is taken to be equal to {{math|φ(''n'')}}."
::::--[[Special:Contributions/50.53.35.240|50.53.35.240]] ([[User talk:50.53.35.240|talk]]) 13:37, 25 October 2014 (UTC)
::::: No. It's the modification that must be explained, not the other way around. Provided a footnote on this matter is justified, it should clearly state that Mathematica very recently (when?) decided to extend to n<1 the phi function, introduced by Euler in 1763, and explain why. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 15:24, 25 October 2014 (UTC)
::::::By [[WP:NPOV]], if we describe the specificities of Mathematica implementation of the function, we must do the same for the other software that implement it, which include at least [[Maple (software)|Maple]], [[Magma (computer algebra system)|Magma]], [[GP/PARI]], [[GAP (software)|GAP]], and certainly also [[Macsyma]], [[Maxima (software)|Maxima]], [[Sage (mathematics software)|Sage]]. For the record, in Maple, the function is called Numtheory[phi]. A section about the various implementations could be written, but not really useful, as the implementation of this function is an easy exercise for a beginner in [[computer algebra]]. Moreover such a section could be based only on [[WP:PRIMARY|primary sources]], and therefore would be [[WP:OR|original synthesis]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:44, 25 October 2014 (UTC)
:::::::The Maple documentation for [http://www.maplesoft.com/support/help/Maple/view.aspx?path=numtheory%2fphi numtheory[phi<nowiki>]</nowiki>] says that it accepts an integer as a parameter, but it doesn't say what the function returns for {{math|''n'' < 1}}. You have a point re [[WP:NPOV]], but neither of you have yet provided a [[wp:rs|sourced]] explanation for why {{math|φ(''n'')}} is not defined for {{math|''n'' < 1}}. BTW, Euler wrote in [[Latin]] ([http://gallica.bnf.fr/ark:/12148/bpt6k6952c/f571.image link] from the article). --[[Special:Contributions/50.53.34.31|50.53.34.31]] ([[User talk:50.53.34.31|talk]]) 05:31, 26 October 2014 (UTC)
::::::::[http://www.sagemath.org/doc/reference/misc/sage/rings/arith.html?highlight=euler_phi Sage]: "Notice that euler_phi is defined to be 0 on negative numbers and 0." --[[Special:Contributions/50.53.34.31|50.53.34.31]] ([[User talk:50.53.34.31|talk]]) 06:27, 26 October 2014 (UTC)
::::::::The [[MuPAD]] function [http://www.mathworks.com/help/symbolic/mupad_ref/numlib-phi.html numlib::phi(n)] accepts an argument that is an "Integer not equal to zero". Example 1 shows that numlib::phi(-7) returns 6. --[[Special:Contributions/50.53.40.60|50.53.40.60]] ([[User talk:50.53.40.60|talk]]) 11:53, 26 October 2014 (UTC)
::::::::The documentation for the [[Maxima (software)|Maxima]] function [http://maxima.sourceforge.net/docs/manual/maxima_29.html#IDX1330 "totient"] does not say what it returns for 0 or negative integers, but I installed it and found that it does what [[Mathematica]] does:
<pre style="margin-left: 16em;">
(%i2) totient(16);
(%o2) 8
(%i3) totient(-16);
(%o3) 8
(%i4) totient(0);
(%o4) 0
</pre>
::::::::--[[Special:Contributions/50.53.40.60|50.53.40.60]] ([[User talk:50.53.40.60|talk]]) 16:08, 26 October 2014 (UTC)
:::::::::The Maxima [[source code]] can be downloaded [http://maxima.sourceforge.net/download.html here]. The "totient" function is implemented in <code>maxima-5.34.1/src/numth.lisp</code>. Unfortunately, there are no explanatory comments, but the implementation is definitely not a [[Bug (software)|bug]]. In particular, the function takes the [[absolute value]] of its argument: {{nowrap|<code>(setq n (abs n))</code>}}. --[[Special:Contributions/50.53.40.60|50.53.40.60]] ([[User talk:50.53.40.60|talk]]) 16:53, 26 October 2014 (UTC)
:::::::::{{tq|"[Maxima] does what Mathematica does"}}
:::::::::[[Mathematica]] and [[Maxima (software)|Maxima]] also agree that [[Greatest common divisor|gcd]](0,0) is 0. ([https://www.wolframalpha.com/input/?i=gcd%280%2C0%29 link] to [[WolframAlpha]])
:::::::::--[[Special:Contributions/50.53.60.41|50.53.60.41]] ([[User talk:50.53.60.41|talk]]) 09:34, 27 October 2014 (UTC)
: "Euler wrote in Latin": wow! what a scoop! So you've been insisting on changing things on this article without knowing the first thing on Euler, and clearly very little on the phi function itself. I am not going to discuss this matter any further with you and your numerous ip avatars. The only sensible thing you wrote here is that I am wasting my time: I will not waste it any more. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 11:04, 26 October 2014 (UTC)
== Alternate way to find Euler's Totient Function of Odd Composite ==
Let 'N' be any odd composite.
Find semi-prime Sp = uv such that
(2u+1)(2v+1) <= N <= 3(2Sp+1). or N <= (2u+1)(2v+1) < 3(2Sp+1)
For example: let N = 65 then required semi-prime is 14 = 2*7 such that
65 < (2*2 + 1)(2*7 +1) < 3*(2*14 + 1) i.e., 65 < 5*15 < 3*29 i.e., 65 < 75 < 87
Note: Finding Semi-prime Sp will be optimized through Trial & Error
Let Lsp be any semiprime nearer and lesser than Sp compared to other.
Then Euler's Totient function Phi(N) = 4*(Sp - k) where 0 <= k <= (Sp - Lsp).
For example: for 65 = 5*13 Phi(65) = 4*12 = 48 = 4*(14-2) Look above example Sp = 14, Lsp = 9, Sp - Lsp = 14-9 = 5;
here we saw that 0 < k < 5 <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Yourskadhir|Yourskadhir]] ([[User talk:Yourskadhir|talk]] • [[Special:Contributions/Yourskadhir|contribs]]) 06:29, 5 March 2015 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:Even if this were accurate and sourced, it still wouldn't seem helpful. — [[User:Arthur Rubin|Arthur Rubin]] [[User talk:Arthur Rubin|(talk)]] 01:56, 11 March 2015 (UTC)
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