Talk:Prime number/Draft

Prime number

Current talk page

Archive 1, Archive 2

Approval area

Commented out V1 To approve tag

Creating Prime number/Draft David Tribe 20:04, 6 May 2007 (CDT) Approved and locked nominated version Approved V 1. About to move talk page of approved to Talk/Draft David Tribe 20:16, 6 May 2007 (CDT)

Approval date for v 1.1 arrives without objection. I see one editor nominating who has only made a small grammar change with a second editor approving as well. I see no dissentions. This article can be re-approved using the Individual Editor approval rules. There are two edits after the approval date shown that appear to be important content issues. Approval Editor okays approval of these two edits. Approval commencing. --Matt Innis (Talk) 18:53, 10 May 2007 (CDT)

---

Article Checklist for "Prime number/Draft"

Workgroup category or categories	Mathematics Workgroup [Categories OK]
Article status	Approved article: approved by editor(s) according to our process
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Jitse Niesen 04:59, 16 May 2007 (CDT)

To learn how to fill out this checklist, please see CZ:The Article Checklist.

---

STOP!! Michael, you cannot approve an article that you have authored BY YOURSELF. The changes between the approved version that is up and the one you are nominated are largely your own changes as author. In that case, You need THREE EDITORS in MATH to do that. So- all 3 have to put up an approval for TODAY. sorry- but as you are the only editor who has worked on the changes since approval as author, you certainly cannot nominate the changed article for new approval. If there are copyedits, then either one of the two nominating editors can contact me. You were not a nominating editor.Please contact the other editors and work together.[It is true that ANOTHER math editor-solo- who has NOT authored in this revision could nominate the revision for approval. Basically, an editor cannot push through approval for something that his or her own work. Greg Martin could nominate for the first approval because he edited a developed article written by others, and Jitse could second because it was a second.If another editor nominates this revision for approval- you could second . )Nancy Sculerati 20:53, 8 May 2007 (CDT)

Mike, not that she needs my support here, but Nancy is correct. If you have contributed significantly to an article, you must first have agreement from two other math editors before it can be nominated for approval by any of you. Please review Approval Process. --Larry Sanger 21:07, 8 May 2007 (CDT)

It's rather vague what "contributed significantly" means. For instance, compare with Nancy's comment dated 22:59, 29 April 2007 on this page. Greg Martin's changes were bigger than Michael's. As I see it, Michael's changes were more than copyediting, but this is a matter of definition, and it is possible to see them as mere copyediting.

A bigger problem in my mind is that the date-of-approve was set to be the date that the ToApprove template was actually added. CZ:Approval Process says that there should at least be 24 hours in between, and that makes sense to me. People should have time to review the article. We're all learning the process as we go along so things will go wrong sometimes.

Anyway, as it happens, I support the changes that Michael made so I put the ToApprove template back up. I don't think I'm considered to be an author in this case: I wrote some parts, but they were approved in V1.0, and my only change since then was definitely a copyedit. I think a short period (one day) is justified because the changes are not big and quite important; the approved version is definitely misleading. -- Jitse Niesen 21:25, 8 May 2007 (CDT)

Actually, I don't think the mention of Euler's result that the sum of the reciprocals of the primes diverges can be considered a "copy edit". But the \scriptstyle changes certainly could be.

I do think it is important to alter the very misleading statement about "unique factorization" in the currently approved version. Michael Hardy 22:09, 8 May 2007 (CDT)

Michael, if you are happy with the version that Jitse has nominated above- add your name to the template. That's the way to show your approval as editor. If you are not happy with it - if that version is not satisfactory- remove the template. That's how you can show that you -as a math editor- feel so strongly that the version being nominated is inadequate for approval. We are all pretty awkward at this (me especially) but we do manage to make the approvals process work. This way- as long as that template is there -on May 10- a constable will approve that version. Nancy Sculerati 22:35, 8 May 2007 (CDT)

Highly misleading phrase

The approved version of this article says:

in fact, it factors completely into prime numbers, due to unique factorization

That is very misleading at best. It implies that uniqueness rather than existence of the factorization is what entails that a number factors completely into primes. That is clearly false. Even in structures within which factorization is not unique at all, elements still factor completely into primes. Possibly the most well-known example is the ring of integers with a square root of −5 adjoined. Michael Hardy 11:37, 7 May 2007 (CDT)

That's right. I'm not sure how the argumnt for the existence of a factorization was removed (I'm pretty sure it was there), though I imagine it may have been a casualty of the removal of proofs. But since Z is Euclidean, we really only need use induction on absolute value. Greg Woodhouse 11:47, 7 May 2007 (CDT)

Hmm...Now that I take a second look, the article doesn't really assert that existence follows from uniqueness, but merely notes that the factorization is, in fact, unique (citing another article). That doesn't strike me as being misleading. Greg Woodhouse 12:00, 7 May 2007 (CDT)

What does "factors completely into prime numbers" mean, if not existence? How could "factors completely into prime numbers" follow from uniqueness? Michael Hardy 19:00, 7 May 2007 (CDT)

I think Michael Hardy has a good point. I had raised a similar point previously, and I find the way it's expressed in the approved version somewhat unsatisfactory. The current draft version may be somewhat better but only if the required (clear) statement and proof are actually contained in the unique factorization page; (which it isn't at the moment); it's still not ideal even then, because it creates confusion between the two propositions (existence of a factorization, and uniqueness of it.) How about this edit: making the link to unique factorization a casual link rather than a "see ...": "On the other hand, every number $N>1$ is divisible by some prime (in fact, it factors completely into prime numbers)." Again, the required statement and proof would have to be included on the unique factorization page. My preference, actually, is to have the complete proof of an infinitude of primes contained on the prime page, as I argued above. --Catherine Woodgold 18:35, 7 May 2007 (CDT)

That would be my preference, too. Originally, I tried to prove everything (though I hope I didn't fall into a boring Bourbaki style, as Sébastien would say), but there semed to be a strong consensus in favor of moving the proofs out of the article. In this case, it doesn't seem to be that the proof itself is all that daunting. Since a proper divisor must be greater than 1, if n is not irreducible, then it must have a proper divisor that is smaller in absolute value. Repeating this process, we can factor n as a product of irreducibles. Now, the point of contention, and what is not so obvious is that irreducible elements are also prime. This is so because $\mathbb {Z}$ is a PID (proved using the division algorithm). Certainly, we can prove that irreducibles are prime without saying anything about principal ideal domains, but it is true that this is a subtle point that appears almost immediately, and it really can't be avoided except by asking the reader to take this result on faith (at least temporarily). I'm certainly open to suggestions as to how this difficulty might be avoided. Greg Woodhouse 19:06, 7 May 2007 (CDT)

Easy. The definition of prime given in the first sentence is "A prime number is a whole number that can be evenly divided by exactly two numbers, namely 1 and itself." I think that's what you mean by irreducible; anyway, either the number is prime, or it is divisible by another number (which evidently must be smaller than itself). --Catherine Woodgold 19:14, 7 May 2007 (CDT)

OK, I've found a concise way to complete the proof without referring to unique factorization or to another page, and I've boldly edited it into the draft: "(for example, its largest divisor greater than 1 must be a prime)". Details that the reader has to fill in (if that one is astute enough to realize that they need to be filled) are that the number N must be greater than 1; that it must have at least one divisor greater than 1 (because it divides itself); and that if a number divides a divisor of N then it must also divide N (by associativity of multiplication). I think it's OK to gloss over these details. I hope the rest of you will feel free to change it again if necessary. --Catherine Woodgold 07:26, 8 May 2007 (CDT)

Unfortunately, that doesn't quite work (consider the case of 8). I went in an added a different argument (which you are welcome to revert), but now it occurs to me that perhaps you meant the smallest proper divisor must be prime. Greg Woodhouse 08:21, 8 May 2007 (CDT)

Right: I meant "smallest". It now says this: " (To see why, let q be the smallest divisor of N; any proper divisor of q would be a smaller divisor of N, so q must be prime.)" which is pretty much what I meant, but I think my version is easier for the reader to follow -- my version doesn't seem to be adding more steps. Also, I mention "greater than 1" which is left out in that version, making it not quite rigourous. I suggest changing to this: "(for example, its smallest divisor greater than 1 must be a prime)". --Catherine Woodgold 20:20, 8 May 2007 (CDT)

I edited in my suggestion above. There are two problems with the current nominated version. First of all, it is incorrect. It says "To see why, let q be the smallest divisor of N; any proper divisor of q would be a smaller divisor of N, so q must be prime." The smallest divisor of N is 1. We can implicitly assume that $N>1$, but we can't say "smallest divisor" when we mean "smallest divisor greater than 1" and claim the proof is correct. The other problem is this: Greg Martin has called this proof "one of the few rigorous mathematical proofs totally accessible to a layperson." Let's not mess that up. The current nominated version uses a quick little implicit, abbreviated proof-by-contradiction in this part, making it a proof-by-contradiction within a proof-by-contradiction. I don't think this is accessible to someone who is just being introduced to proof-by-contradiction for the first time. I think the concise version "for example, its smallest divisor greater than 1 must be a prime" is accessible. --Catherine Woodgold 07:30, 9 May 2007 (CDT)

Good point that we have to add "greater than 1".

I'm not very happy with the way the footnote is formulated. I moved some parts around because I found the phrase "To see why $N$ has a smallest divisor greater than 1, call it $q$, which must be a prime:" too complicated. The word "then" in the last sentence seems out of place. I don't like having a remark in a footnote in a parenthetical remark; it seems a bit too much. Perhaps it's better to have it in the main text. After "We conclude that there are infinitely many prime numbers", we something like: "Actually, if we study the proof carefully, we notice that there is something missing". It's a way to show what rigour means. On the other hand, this may be better in an article about mathematical proof. Finally, I think that the proof that Catherine outlines in her 07:43, 5 May 2007 (CDT) remark.

But this all requires more thought and discussion. Given the time pressure (and the fact that it's past midnight here), it's probably best to focus on having no mistakes.

Oh yes, I updated the template so that the current version is nominated. -- Jitse Niesen 10:19, 9 May 2007 (CDT)

Historical remark regarding 1

The introductory material says that one time mathematicians often did consider 1 a prime. I don't necessarily doubt this, but I've never heard this claim before, either. Does anyone have a reference? In any case, I think prime (as opposed to irreducible) is a relatively modern concept. Greg Woodhouse 11:55, 7 May 2007 (CDT)

MathWorld says "the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909; Lehmer 1914; Hardy and Wright 1979, p. 11; Gardner 1984, pp. 86-87; Sloane and Plouffe 1995, p. 33; Hardy 1999, p. 46)". Fredrik Johansson 12:00, 7 May 2007 (CDT)

Okay, thanks. Maybe we can include a reference or two into the article. This is something that makes sense historically, but is likely to be something of a surprise to a modern reader. On second thought, maybe it would be preferably to say something along the lines of, "the modern definition of prime number requires that primes be > 1". I don't think I want to get too involved in discussing why 1 is not considered a prime, but neither do I want to give undue prominence to the idea of older mathematicians thinking of 1 as a prime, unless of course, we really want to grapple with the concepts of primes vs. irreducibles early on, and that, too, is problematic. Greg Woodhouse 19:20, 7 May 2007 (CDT)

Could it be that some ancient Greeks did not consider 1 a prime number for the simple reason that they did not consider 1 a number? Michael Hardy 18:56, 7 May 2007 (CDT)

I have no clue. As I said before, the history of mathematics isn't my strong point (or even close, really). Greg Woodhouse 19:20, 7 May 2007 (CDT)

Huh?

This has been added to the "draft":

...

consider that if we divide there arer any divisors other than the number itself and 1,

...

"there arer"? I'm trying to figure out what this sentence means, but I'm stumped. Michael Hardy 15:13, 8 May 2007 (CDT)

Ugh! I'm sure I'm responsible for that. I like Catherine's suggested proof (take a minimal divisor) better, anyway, so rather than try to reconstruct the garbled text, I've replaced it with Catherine's proposal. I just hope I got that right! Greg Woodhouse 17:20, 8 May 2007 (CDT)

Scriptstyle?

I thought the more-or-less consensus on the forum was that we should not put "scriptstyle" in a lot of the TeX code to make it look nicer; that that's not what "scriptstyle" is for. My own opinion is that it's better to just put math tags and leave it up to the browser to decide how to display it. I think it makes sense to decide these things by discussion and come up with guidelines to be used on all the math pages. --Catherine Woodgold 20:24, 8 May 2007 (CDT)

That was my understanding, too. I don't know much about the MediaWiki software, but I wonder how hard it would be to create a new tag, say <imath>...</imath> for inline formulas. We could use the new tag for formulas that are displayed inline, and that way the authors won't have to be concerned with the method actually used to acvhieve (some degree of) consistency in font size. To tell you the truth, the single most frustrating thing about this approval process was the controversy over \scriptstyle - one person woul come along and say "put it in", and then someone else would say "take it out". I wasn't a happy camper. Now, personally, I come down on the "use \scriptstyle for its intended purpose" side of the argument, but more importantly, I don't want to be always going back and forth. On top of that, typng isn't really that easy for me, particularly with my left hand, so having to type \scriptstyle over and over, when someone else likely to come along and take it all out isn't exactly fun. Greg Woodhouse 20:45, 8 May 2007 (CDT)

I think this is an issue that the mathematics workgroup--meaning, mathematics editors--need to deliberate about and then ACTUALLY VOTE on. See if you can get a quorum to discuss and vote, so that a decision can be said to be made and to have credibility.

I am in favor of using the best tool for the job. If that means using tools in ways that aren't their intended purpose, big deal. Sorry, call me a philistine. I never cared about how H1-H6 were intended to be used in HTML, either. I greatly prefer the aesthetics of the use of \scriptstyle in my browsers of choice. If most browsers (IE and Firefox notably) don't display plain TeX properly, and if this problem isn't going to be fixed anytime sooner, I think we ought to be pragmatic. I realize that this is asking a lot for certain mathematicians, who prefer simple and elegant solutions over messy and pragmatic ones, but that's my opinion, anyway.

This is just my own nonbinding opinion, and I think the decision ought to be left up to the mathematics workgroup. --Larry Sanger 20:54, 8 May 2007 (CDT)

P.S. Why not use cz-math (the mailing list) to ask for opinions? --Larry Sanger 20:56, 8 May 2007 (CDT)

P.P.S. Can we please have a volunteer who will corral our math editors and extract opinions/votes on the "scriptstyle" issue, and count votes? --Larry Sanger 21:04, 8 May 2007 (CDT)

Nobody has proposed \scriptstyle for "displayed" TeX, but only for "inline" TeX in those cases where the font looks ridiculous when \scriptstyle is not used. Please see CZ talk:Mathematics Workgroup, where some discussion has been going on for a few weeks. Michael Hardy 13:26, 9 May 2007 (CDT)

I've been bold and started an article called CZ:Formatting_mathematics, the purpose of which is to gather together all debates and policies related to formatting mathematics in Citizendium articles. I've put enough content there to suggest the format I have in mind: one main article with the statements of policies, proposed policies, and issues under discussion; and for each item, a subsidiary article with a detailed discussion of the issue, so that people will know why the policies that are (eventually) decided upon are the way they are (I fleshed out one such discussion article to give a sense of what I have in mind). I hope that this creation has value for our workgroup! If so, please feel free to develop the skeleton I put there. - Greg Martin 16:26, 10 May 2007 (CDT)

APPROVED Version 1.1

Congratulations on V1.1! --Matt Innis (Talk) 19:35, 10 May 2007 (CDT)

Discussion for Version 1.1 stopped here. Please continue further discussion under this break.

First paragraph

It says, " In particular, a prime number p cannot be factored as the product of two numbers $\scriptstyle p=a\times b$". This is not true in general. It should say "two whole numbers" or "two integers" rather than "two numbers". --Catherine Woodgold 07:43, 11 May 2007 (CDT)

The default interpretation of "number" is "whole number". I'd say that's true in general even, but certainly in the context of this article - and in the context of factorization, which is a notion defined solely in terms of integers - "number" seems all right to me. Furthermore, "whole number" was used the first time it occurred in the opening paragraph, further setting the default in the reader's mind.

We definitely want to get it exactly right ... but we who are close to the subject might be surprised how much a word like "integers" can present a terminological barrier to non-specialists. (Not to mention we'd have to say "positive integers" anyway, since integers can be zero or negative.) - Greg Martin 11:55, 11 May 2007 (CDT)

How about deleting "whole" from the first sentence, then appending at the end of the paragraph something like "(In this context, "number" is understood to mean "whole number".)"? In a way, saying "whole number" and then "number" is even worse than just saying "number" every time: it establishes that there are different kinds of numbers and then later doesn't specify which. Well, if you think in terms of ordinary language, "number" could be seen as an implied abbreviation ("anaphor"?) for the phrase "whole number" which was already used; but not if you think like a mathematician. And you're right, I was wrong about saying "integer" because $5=-5\times -1.$, which would be OK I suppose if primes were defined differently. --Catherine Woodgold 18:33, 11 May 2007 (CDT)

Other possibilities: beginning with "Within the whole numbers, a number that can be evenly divided by exactly two numbers, namely 1 and itself, it is called a prime number." or just moving the word "whole": "A prime number is a number that can be evenly divided by exactly two whole numbers, namely 1 and itself."

In the third sentence: "In particular" seems out of context to me. I would just delete it. --Catherine Woodgold 08:23, 12 May 2007 (CDT)

I went for moving the word "whole". It was the only suggestion which did not strike me as exaggeratedly rigour. We all know that there is a place for rigour in mathematics, but I'd like the first sentence be as natural as possible.

I deleted "in particular" as suggested. -- Jitse Niesen 04:59, 16 May 2007 (CDT)

Proof by contradiction

I think the proof by contradiction could be explained better. It says "This contradiction is irreconcilable unless we admit that our set of prime numbers was not complete after all. Since no finite set of prime numbers can be complete, we conclude that there really are infinitely many prime numbers." To me, this is mixing up two different kinds of proofs.

Proof number 1: Take any set of prime numbers, and show that there is another one that is not a member of the set. Since for any finite set another one can be found that is not in the set, there must be an infinite number of them. (In other words, when you try to enumerate them they keep going.)

Proof number 2: Proof by contradiction. Assume there is a finite number of prime numbers. A contradiction ensues. Since we got a contradiction after making a certain assumption, that assumption must be false. Therefore there is not a finite number of prime numbers -- it must be an infinite number.

I think we should choose one or the other, not mix up these two different kinds of proofs. To me, a contradiction is a contradiction and that's that: we don't "reconcile" it.

Instead, we could say:

We have arrived at a contradiction after supposing that the set of prime numbers is finite. Therefore that supposition must be false, and the number of prime numbers must be infinite. This completes the proof.

--Catherine Woodgold 19:07, 11 May 2007 (CDT)

Euclid's proof was by contradiction. I think the one you've labelled "Proof number 1" is clearer, since it will not mislead the reader into the false conclusion that 1 + the product of the first several primes is necessarily prime. Note that 2×3×5×7×11×13 + 1 is composite, since it is 59×509. If Euclid weren't such a big name, maybe "Proof number 1" would be seen more often. Michael Hardy 20:38, 11 May 2007 (CDT)

Based on the translation of Euclid's proof at http://aleph0.clarku.edu/%7Edjoyce/java/elements/bookIX/propIX20.html , it seems to me that Euclid's proof is in fact closer to "Proof number 1". -- Jitse Niesen 21:40, 11 May 2007 (CDT)

Euclid does not use a proof by contradiction for the proof as a whole: he uses it for the part we skimmed over by saying that multiples of primes are never "next to each other". I agree, his proof as a whole looks more like proof number 1. Also, I think this theorem naturally lends itself to being done by proof number 1. If the contradiction were something irrelevant, such as "$1=0$", then it would make sense to do a proof by contradiction. But what is being proved in particular is that there is always one more prime, so that lends itself to going directly from there to saying that the number of primes is infinite.

I suggest replacing the first two sentences of the proof with:

Euclid proved that for any finite set of prime numbers, there is always another prime number which is not in that set. Choose any finite set of prime numbers $\scriptstyle \{p_{1},p_{2},p_{3},\ldots ,p_{n}\}$.

And the end of the proof, replace "This contradiction is irreconcilable unless..." with "Therefore...".

I don't see why proof number 1 or number 2 would be any more or less likely to be misunderstood to mean that $N+1$ is prime, though. That misunderstanding happens (or doesn't) in the middle of the proof, which can be the same in both cases. --Catherine Woodgold 07:41, 12 May 2007 (CDT)

That looks like a sound suggestion, so I implemented it. I didn't quite understand what you wanted to do at the end of the proof, so I improvised there. -- Jitse Niesen 08:05, 12 May 2007 (CDT)

Thanks. Your "This shows that..." works better then my "Therefore...". --Catherine Woodgold 08:31, 12 May 2007 (CDT)

Too many hands

I kind of like the "on the one hand ... on the other" idiom here, but you've got to stop at two. We only have two hands. :-) Greg Woodhouse 19:50, 12 May 2007 (CDT)