Talk:Euler's totient function

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. -Waylonflinn 22:38, 28 December 2006 (UTC)

How to pronounce totient?—Tokek (talk) 08:15, 23 June 2005 (UTC)

Totient rhymes with division's quotient. —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. 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.—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. Bubba73 18:01, 23 June 2005 (UTC)

Noted. —Tokek (talk) 19:06, 23 June 2005 (UTC)

Doesn't totient(n) always equal a positive integer? Railgun 16:59, 26 June 2006 (UTC)

What does the phrase "randomly large n" mean? Should that be "general n"? 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? $\sum _{d|n}f(d)\;$   and   $\prod _{d|n}f(d)\;$   mean that the sum or product is over all positive divisors of n, including 1 and n. E.g., if n = 12,

$\prod _{d|12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).\$

p|n also needs explanation as an index = all prime divisors of n. And possibly the Fourier Transform notation... Thanks?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:

$\prod _{p|n}(1-{\frac {1}{p}})=(1-{\frac {1}{1}})\times \cdots \times (1-{\frac {1}{n}})=0.\$

That is pretty clearly incorrect, but it is what a literal reading of the notation description would lead one to believe.— Preceding unsigned comment added by 128.138.65.99 (talk • contribs) 15:10, 30 May 2014‎

The article says "where the product is over the distinct prime numbers dividing n". As 1 is not a prime number, $(1-{\frac {1}{1}})$ cannot appear in the product, and the product is never zero. Similarly, the factor $(1-{\frac {1}{n}})$, may appear only if n is prime. In this case, it is the only factor.

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… 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. —David Eppstein (talk) 23:29, 7 July 2014 (UTC)

Definition of φ(0)

The edit summary for this reversion says: "The function is not defined for n=0." That statement should be in the article and sourced. See Hardy & Wright: 5.5 Euler's function φ(m).

The edit summary for 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 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."

(Weisstein, Eric W. "Totient Function". MathWorld.)

--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. 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. — D.Lazard (talk) 12:28, 24 October 2014 (UTC)

1. The article should explicitly state that φ(0) is undefined by Hardy & Wright and any other number theory sources you care to cite. The table should conform to that.
2. A bug is a mistake. The Mathematica convention is not a bug. The Mathematica documentation for 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.
3. 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.

--50.53.60.172 (talk) 13:20, 24 October 2014 (UTC)

Clarification: Hardy & Wright do not explicitly state that φ(0) is undefined. In their definition of φ(m), the domain of φ is implied by their inequality:

• $0<n\leq m$ (§ 5.5, p. 52)

--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 explicitly states that φ(0) is undefined by various cited sources. --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. Sapphorain (talk) 16:40, 24 October 2014 (UTC)

"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 verifiable and reliable sources, so it would be more constructive if you provided some sources making your point.

And the article is biased, because it does not say what 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." (link)

Do you know what that means?

For your convenience, here are links to the Mathematica documentation for EulerPhi and FactorInteger.

--50.53.60.172 (talk) 17:39, 24 October 2014 (UTC)

Yes, indeed, I think I have been wasting my time. Sapphorain (talk) 21:07, 24 October 2014 (UTC)

'..."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 EulerPhi[-60] is 16.

--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. Sapphorain (talk) 08:01, 25 October 2014 (UTC)

(edit conflict) This simply confirm that Mathematica is not a reliable source. No need of wasting our time for that. D.Lazard (talk) 08:07, 25 October 2014 (UTC)

Sapphorain: "... an extension of the Euler function ..."

That's an excellent way to describe the domain of the software function EulerPhi. Haven't any mathematicians discussed the domain of the mathematical function φ? A footnote about EulerPhi would be fine, if the article clearly stated that mathematicians do not define φ(n) for n < 1 and explained why. (It has something to do with the solution to the equation gcd(n, k) = 1 for n < 1.)

D.Lazard: "... Mathematica is not a reliable source."

The Mathematica documentation is a reliable source for what the software function EulerPhi does. Indeed, the documentation says:

• "φ(-n) is taken to be equal to φ(n)."

--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. 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, Magma, GP/PARI, GAP, and certainly also Macsyma, Maxima, 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 primary sources, and therefore would be original synthesis. D.Lazard (talk) 16:44, 25 October 2014 (UTC)

The Maple documentation for numtheory[phi] says that it accepts an integer as a parameter, but it doesn't say what the function returns for n < 1. You have a point re WP:NPOV, but neither of you have yet provided a sourced explanation for why φ(n) is not defined for n < 1. BTW, Euler wrote in Latin (link from the article). --50.53.34.31 (talk) 05:31, 26 October 2014 (UTC)

Sage: "Notice that euler_phi is defined to be 0 on negative numbers and 0." --50.53.34.31 (talk) 06:27, 26 October 2014 (UTC)

The MuPAD function numlib::phi(n) accepts an argument that is an "Integer not equal to zero". Example 1 shows that numlib::phi(-7) returns 6. --50.53.40.60 (talk) 11:53, 26 October 2014 (UTC)

The documentation for the Maxima function "totient" does not say what it returns for 0 or negative integers, but I installed it and found that it does what Mathematica does:

(%i2) totient(16);
(%o2)                                  8
(%i3) totient(-16);
(%o3)                                  8
(%i4) totient(0);
(%o4)                                  0

--50.53.40.60 (talk) 16:08, 26 October 2014 (UTC)

The Maxima source code can be downloaded here. The "totient" function is implemented in maxima-5.34.1/src/numth.lisp. Unfortunately, there are no explanatory comments, but the implementation is definitely not a bug. In particular, the function takes the absolute value of its argument: (setq n (abs n)). --50.53.40.60 (talk) 16:53, 26 October 2014 (UTC)

"[Maxima] does what Mathematica does"

Mathematica and Maxima also agree that gcd(0,0) is 0. (link to WolframAlpha)

--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. 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 — Preceding unsigned comment added by Yourskadhir (talk • contribs) 06:29, 5 March 2015 (UTC)

Even if this were accurate and sourced, it still wouldn't seem helpful. — Arthur Rubin (talk) 01:56, 11 March 2015 (UTC)

The name of the function

I am an American number theorist, and I cannot find a single book in my shelves that calls the Euler phi function the "totient" or "Euler totient" function. It is in all cases referred to as the Euler phi function. I think for a function as well-used as this one, especially by undergraduates, we should use standard terminology. It is hard to cite all the references -- please look at any number theory book on your shelf. I am curious about what other readers think.Jaatex (talk) 19:19, 11 April 2015 (UTC)

I am an American number theorist, and in no reference on my shelf is the Euler phi function called the totient function or the Euler totient function. With a function as well-used as this, especially by undergraduates, shouldn't we use the standard name? I'm curious how others feel. Jaatex (talk) 19:27, 11 April 2015 (UTC)

I am also a number theorist, and have very often heard the phi function called the "totient function", or "Euler totient function". The term was apparently invented by Sylvester. It is mentioned by Niven and Zuckerman (3rd edition, p.22). In the "Handbook of estimates in the theory of numbers" (B. Spearman, K.S. Williams, Carleton Mathematical Lecture Note 14, 1975), on the definition p. 4, phi is called "Euler's totient function". If you search on MathSciNet, you will find that the number of articles containing in their titles "Euler totient function" is 34 (73 for "Euler phi function"), and that the number of referee's reports citing "Euler totient function" is 386 (225 for "Euler phi function"). (Searching for just "totient function" is misleading: there are many other totient functions). Sapphorain (talk) 20:57, 11 April 2015 (UTC)

Thanks for the research. As you found, two-to-one in titles it is referred to as the Euler phi function. My bookshelf research: Ireland and Rosen, "A Classical Introduction to Modern Number Theory" and Pierre Samuel (translated from French), "Algebraic Theory of Numbers"; Kenneth H. Rosen "Elementary Number Theory"; Joseph H. Silverman, "Number Theory"; David M. Burton, "Elementary Number Theory" (the only reference where "totient" is mentioned: "The function \phi is usually called the \it {Euler phi-function} (sometimes, the \it indicator or \it totient)"); Serge Lang, "Algebra"; and "Andre Weil, "Number Theory, An approach through history...". In the above references, totient is neither in the index nor the text (except as noted in Burton, where it is not in the index). I understand that totient function terminology is sometimes used, and that it is historical. My point is that it is not commonly used. If a student tries to look up the totient function in a standard text, they will find nothing. I'm not sure why totient was chosen for the title of this article. I would like to change it to the more common name. Jaatex (talk) 14:39, 15 April 2015 (UTC)

In the past, this page was named "Euler's phi function". It is not clear when the name has changed, but the first line has been changed in 2003, apparently without any discussion, by this edit [1]. In any case, as both Phi function and Euler's phi function redirect to this article, the reader searching for this article will find it easily. Personally, I have no opinion on the best title nor for the name that should be used in the article (as far as both names appear in the first sentence). D.Lazard (talk) 15:23, 15 April 2015 (UTC)

Thank you for this information. This is my first attempt to edit a Wikipedia page. I see the edit from 2003, thanks for pointing it out. I do not see why the change was made. How was the change made without discussion? How does one go about getting it changed back? Do people vote?  :-) Jaatex (talk) 15:54, 15 April 2015 (UTC)

In Wikipedia, decisions are taken by consensus, as described in details in WP:CON. However, few people have given their opinion. One could start a WP:request for comments. But, as nobody disagrees formally with you, and your arguments are convincing, I have self-reverted my revert of your edits. The change of title (called move in WP) seems less important than editing the content, because the title you suggest exists as a WP:redirect. For technical reasons, it may be done (in this case) only through a WP:move request. D.Lazard (talk) 19:12, 15 April 2015 (UTC)

Totient numbers

This section contains there are infinitely many nontotients, and indeed every odd number has an even multiple which is a nontotient, the word "even" being added by a recent edit. In both versions, this sentence is a nonsense, as every odd number greater than 1 is a nontotient; thus no need to consider multiples. I guess that the correct assertion should be every totient has a multiple (by an odd number) that is a nontotient. However this needs to be checked on the source. D.Lazard (talk) 09:59, 25 April 2015 (UTC)

As you say, odd numbers greater than 1 are trivially nontotients, so what is of interest is the existence of even nontotients. I imagine the previous writer meant "there are infinitely many even nontotients, and indeed..." but just forgot the "even". I've checked the paper and in fact it proves that any number (even or odd) has a multiple which is a nontotient. If n is odd then the nontotient multiple of 2n gives an even nontotient multiple of n, so this is equivalent to saying any number has an even nontotient multiple. I'll make those changes. Especially Lime (talk) 08:54, 20 May 2016 (UTC)

Unexplained revert of style improvement

My edit in Euler's totient function#Euler's product formula has been reverted by an IP user, without any explanation. My edit consisted in

• Removing, per MOS:HEADINGS the redundant reference (through a formula) to the article title.
• Replacing a heading consisting of a technical formula by a less technical phrase (the formula was redundant, as reproduced in the body
• Avoiding the confusing term "modulo-and-coprime", which is nowhere defined in Wikipedia
• Linking coprime

All are style improvements that does affect in any way the content of the article. As I cannot find any valid reason for rejecting these edits, I have restored them. Please, if I have missed something, please discuss here before a second revert. D.Lazard (talk) 15:52, 13 January 2017 (UTC)

Not Empty Set is that Correct - Puzzling

Empty set says (and not related in any way to the Greek letter Φ), but inspired by the letter Ø in the Norwegian and Danish alphabets.

My last month Mathematics Today Magazine from mtG has a sentence,-

"Empty set is subset of every set and every set is subset of itself. We denote by it by Φ or {}"

is that phi is small case, couldn't categorize on that font whether it is lower or upper alphabet.

And however, Φ(n) -> Phi(n) -> Euler's totient function

Maybe we can see through Article, and make reference of Empty set if it is relevance to do that as per the sentence made by mtG

—Dev Anand Sadasivam^t@lk 02:26, 30 June 2018 (UTC)

Letter Phi in mathematics

(From my talk page, + answer, Sapphorain (talk) 22:23, 26 December 2018 (UTC))

Hello, regarding your recent revert of my edit, could you provide me an example of some respectable mathematical writing where the author would use the glyphs "φ" and "ϕ" as different variables (to denote two different things)? --Alexey Muranov (talk) 22:08, 26 December 2018 (UTC)

No I cannot, and I am not interested in finding one. But it is not the point. Two different ways of writing the same letter have been used in many instances to denote different objects. So the present precision is quite legitimate and there is no reason to suppress it. Sapphorain (talk) 22:23, 26 December 2018 (UTC)

I agree with showing the two ways it is commonly written. Bubba73 ^{You talkin' to me?} 23:25, 26 December 2018 (UTC)

Although they both refer to the Greek letter phi, the two are separate symbols. I believe they both should be shown or described. In fact, I have seen some fields of mathematics prefer one symbol to the other.—Anita5192 (talk) 23:28, 26 December 2018 (UTC)