Citizendium - a community developing a quality, comprehensive compendium of knowledge, online and free.
Click here to join and contribute
CZ thanks our previous donors. Donate here. Treasurer's Financial Report

Talk:Prime number/Archive 2

From Citizendium
Jump to: navigation, search


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 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 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 , 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 has a smallest divisor greater than 1, call it , 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)

First paragraph

It says, " In particular, a prime number p cannot be factored as the product of two numbers ". 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 , 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 "", 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 .
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 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)

Book IX, Proposition 20

At this web page we find a translation of Euclid's proof (although as far as I've noticed, it does not say who the translator is):

Prime numbers are more than any assigned multitude of prime numbers.
Let A, B, and C be the assigned prime numbers.
I say that there are more prime numbers than A, B, and C.
Take the least number DE measured by A, B, and C. Add the unit DF to DE.
Then EF is either prime or not.
First, let it be prime. Then the prime numbers A, B, C, and EF have been found which are more than A, B, and C
Next, let EF not be prime. Therefore it is measured by some prime number. Let it be measured by the prime number G.
I say that G is not the same with any of the numbers A, B, and C.
If possible, let it be so.
Now A, B, and C measure DE, therefore G also measures DE. But it also measures EF. Therefore G, being a number, measures the remainder, the unit DF, which is absurd.
Therefore G is not the same with any one of the numbers A, B, and C. And by hypothesis it is prime. Therefore the prime numbers A, B, C, and G have been found which are more than the assigned multitude of A, B, and C.
Therefore, prime numbers are more than any assigned multitude of prime numbers.

Apparently, Euclid did not actually start from the assumption that his initial finite set of primes contains ALL primes. Michael Hardy 14:39, 29 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)

Typography in the Sieve of Eratosthenes

An apparent(?) error in this article was pointed out in the forums. In one sense, th error was only apparent, as the markup is correct, but in another sense, it is quite real, as the reader cannot tell the difference. This is my reply from the forum:

I did take a look at the prime number article, and it seems that there is a line through the 4's, too (the markup is <s>4</s>), but it is invisible because of the way the 4 is written. I'll copy this comment over to the talk page.

For reference, the forum posting was http://forum.citizendium.org/index.php/topic,1031.0.html

Greg Woodhouse 14:18, 15 June 2007 (CDT)

I like how the wikipedia article lists the first 30 prime numbers, can we add?

I like how the wikipedia article has the first 30 prime numbers listed. Can we add it? Tom Kelly 14:41, 15 June 2007 (CDT)

Sure. Michael Hardy 17:17, 31 July 2007 (CDT)

A clear error is in the approved version

Per discussion above, the following statement in the approved version is an error:

Euclid's proof is a proof by contradiction.

The draft version corrects it. I think we should replace it as soon as we have an approvable draft. Michael Hardy 17:17, 31 July 2007 (CDT)

... and I've nominated a draft for approval. Michael Hardy 13:55, 1 August 2007 (CDT)
I'm not sure that it is an error. As the article says now, "Euclid proved that for any finite set of prime numbers, there is always another prime number which is not in that set." I think that's an acceptable representation from the text of the Elements, even though the text only shows how, given a set of three prime numbers, to find another one. Then, Euclid concludes that there is an infinitude of primes ("Prime numbers are more than any assigned multitude of prime numbers"). Is there any way other than by contradiction that he could have reached this conclusion? -- Jitse Niesen 07:29, 3 August 2007 (CDT)

He did not reach it by contradiction. That's a historical error that you yourself pointed out on this very page. Michael Hardy 10:09, 3 August 2007 (CDT)

On re-reading the start of the section #Proof by contradiction, I agree and I feel very stupid. -- Jitse Niesen 00:23, 4 August 2007 (CDT)

Comments on latest draft

Some comments on the draft found [[1]]:

  • It opens by stating that A prime number is a number that can be evenly divided by exactly two whole numbers. Since the link is to integer, which explicitly mentions naturals, negatives, and zero, this is incorrect. Whole number should be replaced by natural number or by positive whole number.
  • Same problem occurs later in the first paragraph when it talks about factoring into a product of "two numbers." That should probably be "two positive integers".
  • Same problem in paragraph 3. "Unique factorization of numbers" should be "integers" or "positive integers", perhaps.
  • Under There are infinitely many primes, before This shows that that some prime numbers exist outside of our initial final set, I think it would be clearer if it was stated explicitly that the prime that divides N is at least one prime not in our original list.

Since this was a request for comments, and I am not yet entirely clear how collaboration is supposed to happen in Citizendium (as opposed to Wikipedia), I have not attempted to make such changes. Arturo Magidin 11:58, 2 August 2007 (CDT)

Let's start with the first point. There has in fact been some discussion about it (see the section #First paragraph above) and so I'd like some more input, especially since the first sentence is very important to get right. However, nobody raised the obvious point that we should restrict to positive numbers (perhaps because "whole number" is sometimes used in the sense of "positive integer"), and it didn't cross my mind.
Obviously, we want to make the article, and especially the first sentence, accessible for everybody. "Integer" or "natural number" might be a problem. However, we also want to be correct. I think that "positive whole number" is acceptable for the first sentence, even though some people might perhaps have to read it twice.
By the way, you should feel free to edit the article. Be Bold applies here too. -- Jitse Niesen 07:29, 3 August 2007 (CDT)

Links shouldn't always be only to synonyms. If integer contains an account of the relevant concept, then it may be the most appropriate thing to link to. Michael Hardy 14:22, 3 August 2007 (CDT)

I like "positive whole number". I'm aware that we aim at as easy explanation as possible, but the adjective "positive" is not a very sophisticated word and the sentence reads well. When we have "positive whole number" then the "whole number" may link to "integer" and the meaning is clear; without qualification ("positive") it was somewhat puzzling. Aleksander Stos 11:53, 5 August 2007 (CDT)

APPROVED Version 1.2