Talk:Prime number/Draft: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Michael Hardy
(→‎Book IX, Proposition 20: really a subsection)
imported>Barry R. Smith
(reapprove to fix "error"?)
 
(65 intermediate revisions by 14 users not shown)
Line 1: Line 1:
{{checklist
{{subpages}}
|                abc = Prime number
|                cat1 = Mathematics
|                cat2 =
|                cat3 =
|          cat_check = n
|              status = 0
|        underlinked = y
|            cleanup = y
|                  by = --[[User:Aleksander Stos|Aleksander Stos]] 12:03, 27 May 2007 (CDT)
}}
{{Archive box|auto=long}}
==Approval area==
<!--{{ToApprove
|url = http://en.citizendium.org/wiki?title=Prime_number/Draft&oldid=100099326
|now = 09:59, 9 May 2007 (CDT)
|editor=Jitse Niesen
|editor2=Michael Hardy
|editor3=
|editor4=
|group=Mathematics
|group2=
|group3=
|date =May 10, 2007
}}-->


Commented out V1 To approve tag
==What kind of number==
<!--
The first sentence doesn't specify what kind of numbers we are dealing with. [[User:Andres Luure|Andres Luure]] 22:26, 5 November 2007 (CST)
{{ToApprove|editor=Greg Martin|editor2=Jitse Niesen|url=http://en.citizendium.org/wiki?title=Prime_number&oldid=100097398|group=Mathematics|date=May 6, 2007}}-->


Creating [[Prime number/Draft]] [[User:David Tribe|David Tribe]] 20:04, 6 May 2007 (CDT)
: It says "A prime number is a number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself."  The word "itself" implies it must be a positive whole number.  (But maybe it could be more explicit.) [[User:Michael Hardy|Michael Hardy]] 14:03, 20 December 2007 (CST)
Approved and locked nominated version Approved V 1. About to move talk page of approved to Talk/Draft [[User:David Tribe|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 dissentionsThis 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 issuesApproval Editor okays approval of these two editsApproval commencing. --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 18:53, 10 May 2007 (CDT)
:: And why not be more explicit? It seems much clearer to me, a number theorist even, if you say "a prime number is a positive whole number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself"In the original version, I didn't immediately return to the beginning of the sentence and think, "oh, that IMPLIES that the original number was a positive whole number"If the only concern is that repeating the same phrase twice might be a turn off, let me quote from Strunk and White's "Elements of Style", one of the guides to style that we are supposed to take advice from (see section 19) "The likeness of form enables the reader to recognize more readily the likeneses of content and function...the unskilled writer often violates this principle mistakenly believing in the value of constantly varying the form of expression.  When repeating a statement to emphasize it, the writer may need to vary its formOtherwise, the writer should follow the principle of parallel construction." It seems to me that the repetition in our case is not for emphasis, and after the repetition, it will be firmly fixed in the reader's mind that all numbers begin considered are positive whole numbers.[[User:Barry R. Smith|Barry R. Smith]] 20:40, 29 March 2008 (CDT)


----
I am not sure it is even clear to say "exactly two" when it could be "two and only two." --[[User:Thomas Simmons|Thomas Simmons]] 19:56, 8 November 2007 (CST)


{{checklist
: In mathematics, at least, the phrases "there are exactly two" and "there are precisely two" are understood to express the same statement as "there are two, and only two" (for instance, see the discrete math text I taught out of this past term, or the Wiki page on if and only if).  I have considered these as equivalent for many years, so it is hard for me to put myself in the shoes of someone who might be seeing this for the first time.  The issue that you are concerned with is that someone might accidentally confuse "exactly two" with the idea that it has at least two, but possibly more, positive divisors.  I cannot see how even people with very little mathematical experience would interpret "exactly two" in this manner, the word "exactly" being inserted exactly (hehe) to let you know that this is the precise number. Furthermore, I think it is hard to argue against "exactly two" being the more elegant phrase.  I much prefer the phrasing of the first sentence of the approved article to the first sentence of the current draft.  Any rebuttals?[[User:Barry R. Smith|Barry R. Smith]] 20:40, 29 March 2008 (CDT)
|                abc = prime number
|                cat1 = Mathematics
|                cat2 =
|                cat3 =
|          cat_check = n
|              status = 0
|        underlinked = n
|            cleanup = y
|                  by = [[User:Jitse Niesen|Jitse Niesen]] 04:59, 16 May 2007 (CDT)
}}


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 . )[[User:Nancy Sculerati|Nancy Sculerati]] 20:53, 8 May 2007 (CDT)
:: I agree. Even to naive readers "exactly two" cannot possibly mean "at least two". [[User:J. Noel Chiappa|J. Noel Chiappa]] 22:12, 29 March 2008 (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 [[CZ:Approval Process|Approval Process]]. --[[User:Larry Sanger|Larry Sanger]] 21:07, 8 May 2007 (CDT)
Footnotes versus links to stubs: Until many of the terms are explained with their own articles, the use of footnotes to explain terms and analogies should continue. Otherwise we will have dead links in red letters for a long time to come. This will also mean that the article can reach a broader population as it is written. The chemistry metaphor is another example, that comparisons might be lost on anyone who is not up to speed on freshman level chemistry. So for ease of use and market appeal and just plain educational focus, explaining terms in footnotes would be a good idea. The high school students using CZ today will be the grad students referring to it in future.--[[User:Thomas Simmons|Thomas Simmons]] 19:56, 8 November 2007 (CST)


: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.
== What i miss ==
: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. -- [[User:Jitse Niesen|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.
There are some things, that are not in thearticle:
:
I do think it is important to alter the very misleading statement about "unique factorization" in the currently approved version. [[User:Michael Hardy|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. [[User:Nancy Sculerati|Nancy Sculerati]] 22:35, 8 May 2007 (CDT)
*Prime numbers and Pseudoprimes ([[Fermat pseudoprime]], [[Euler pseudoprime]], [[Carmichael number]], ...)
*Properties of Prime numbers
**p is a Prime number <=> p|(p over n) for 1<n<p
**Prime numbers and Perrin sequence
**Prime numbers aund Lucas sequence
--[[User:Karsten Meyer|arbol01]] 05:04, 1 January 2008 (CST)


== Highly misleading phrase ==
: I don't understand the first comment under "Properties of prime numbers"
: As for prime divisors of elements of those two particular sequences, it seems to me that these are far too specialized to be included in this page, and would be better placed on the "Perrin sequence" and "Lucas sequence" pages separately.  Otherwise, one would need to enumerate ALL named recursively defined sequences, and the divisibility properties in each case.  I would imagine that just this task would encompass many pages in itself.[[User:Barry R. Smith|Barry R. Smith]] 00:20, 30 March 2008 (CDT)


The approved version of this article says:
::**p is a Prime number <=> p|(p over n) for 1<n<p
:: Iff p is a prime number, than p divides (p over n) [Binomialcoeffizient] for every n between 1 and p.


:: in fact, it factors completely into prime numbers, due to unique factorization
:: If p is a prime number, than p divides P<sup>p</sup> and p divides U(P,Q)<sup>p</sup> - P


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 &minus;5 adjoined. [[User:Michael Hardy|Michael Hardy]] 11:37, 7 May 2007 (CDT)
::are properties, that belongs to the Prime numbers. It belongs to the Perrin sequence respectively to the Lucas requence respectively to the binomialcoefficence too. --[[User:Karsten Meyer|Karsten Meyer]] 02:45, 19 May 2008 (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. [[User:Greg Woodhouse|Greg Woodhouse]] 11:47, 7 May 2007 (CDT)
== 1 revisited ==


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. [[User:Greg Woodhouse|Greg Woodhouse]] 12:00, 7 May 2007 (CDT)
Regarding my above comment in "What kind of number" above, I personally feel that the first sentence should read something like, "A prime number is a whole number greater than 1 that can be evenly divided by exactly two positive whole numbers, namely 1 and itself".  It seems that the main argument above against saying a prime must be bigger than 1 from the outset is the need for clarity in the first sentence, but I feel that currently this clarity comes at the price of correctness.  As written, I feel the that first sentence is plain wrong, and I personally wouldn't put my stamp of approval on it.


:: What does "factors completely into prime numbers" mean, if not '''existence'''? How could "factors completely into prime numbers" follow from '''uniqueness'''? [[User:Michael Hardy|Michael Hardy]] 19:00, 7 May 2007 (CDT)
I don't want to sound TOO dismissive.  I didn't just go edit the draft, because I understand that their was some discussion about this above.  Apparently, the status of 1 seems to have been problematic even when the fundamental of arithmetic were laid down in Euclid's "Elements". However, it seems to me that the tone of the approved version suggests that the typical modern "choice" to label 1 as neither prime nor composite is a result of whimsy or chance.  This is a false impression. 


I think Michael Hardy has a good pointI 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 <math>N>1</math> is divisible by ''some'' prime (in fact, it [[Unique factorization|factors completely]] into prime numbers)."'' Again, the required statement and proof would have to be included on the unique factorization pageMy preference, actually, is to have the complete proof of an infinitude of primes contained on the prime page, as I argued above. --[[User:Catherine Woodgold|Catherine Woodgold]] 18:35, 7 May 2007 (CDT)
In a sense, I guess, defining 1 as special can seem as arbitrary as defining 0 factorial to be 1But with the invention of the gamma function and the recognition of its canonical properties, can there be any dispute as to the correct definition of 0 factorial?  Similarly, there are very sound reasons that 1 has been given special status over the last century or so.  The easiest to explain is that the Fundamental Theorem of Arithmetic is just false if 1 is considered prime:  considere, 6 = 2*3 = 2*3*1 -- two different prime factorizations.  (By the way, I also think that the words "Fundamental Theorem of Arithmetic" should appear somewhere on the "prime number" page -- can't remember if I saw it anywhere). A second reason is that with the development of algebraic number theory, the units in algebraic number fields were found to play a very special and important roleWithin the integers, 1 and -1 are the only units, so it is hard to get a feel for the special role they play only within this contextNevertheless, the fact that 1 is the unique multiplicative identity within the integers should make a strong impression(For more about 1, see this website http://mathforum.org/kb/message.jspa?messageID=1379707, and especially the comments by John Conway, a world-renown number theorist.)


: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 <math>\mathbb{Z}</math> 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. [[User:Greg Woodhouse|Greg Woodhouse]] 19:06, 7 May 2007 (CDT)
In summary, although the status of 1 might have fluctuated in the past, I believe the consensus of the vast majority of working mathematicians at present is that it should not be considered prime, and this is reflected in todays high-school textbooks. Furthermore, I do not see any indication that this will change soon. Thus, it seems that the proper definition should make it clear that 1 is not prime from the first sentence. Otherwise, we will be spreading disinformation to those casual learners who wonder, "hmm, I wonder if 1 is a prime", look at the first line of the Citizendium page, and then wander off to tell their friends what they learned.[[User:Barry R. Smith|Barry R. Smith]] 01:31, 30 March 2008 (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). --[[User:Catherine Woodgold|Catherine Woodgold]] 19:14, 7 May 2007 (CDT)
: Dude, you're the expert! I (at least, can't speak for everyone) defer to your clear familiarity. So I'd go for it. Plus to which, your point about the Fundamental Theorem of Arithmetic is good (and so easily understandably by all that it should probably be mentioned in the article as a reason why 1 is not considered by mathematicians as being part of the set of prime numbers, even though by the simplistic definition of 'prime', it seems to be prime). [[User:J. Noel Chiappa|J. Noel Chiappa]] 11:40, 30 March 2008 (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. --[[User:Catherine Woodgold|Catherine Woodgold]] 07:26, 8 May 2007 (CDT)
:Barry, the first sentence currently says "A prime number is a positive whole number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself." I believe this does say that 1 is not prime, just as you want, as the number 1 has only one divisor, namely 1 itself. So I'm not sure what your point is.
:I agree with all the rest you wrote (for what it's worth, as you know of course more number theory than I do). -- [[User:Jitse Niesen|Jitse Niesen]] 16:00, 30 March 2008 (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. [[User:Greg Woodhouse|Greg Woodhouse]] 08:21, 8 May 2007 (CDT)
:: Someone careful and analytic might draw that conclusion, but not all our readers might fit that definition. Baldly saying '1 is not a prime number' is probably what they need. Without in any way intended to be demeaning to them, I am always mindful of that wonderful George Carlin line: "Think of how dumb the average person is - and then realize that half of them are dumber than ''that''."
:: (Adding "different" - as in "two different positive whole numbers" - might make the definition cast-iron, though). But it might still be useful to have a section on 'why 1 is not a prime number'; the point about the Fundamental Theorem of Arithmetic could go there. [[User:J. Noel Chiappa|J. Noel Chiappa]] 23:31, 30 March 2008 (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 stepsAlso, 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)"''. --[[User:Catherine Woodgold|Catherine Woodgold]] 20:20, 8 May 2007 (CDT)
::: Yes, Jitse, it seems that after all of that, I understand 1 okay but I still have trouble counting to 2 :-)Anyway, my own error emphasizes the point that inferring information about the prime number from information presented at the END of the sentence is not my own thought process, and probably not a lot of other people's.  (I suppose if I still kept up my German, I would be used to that sort of thing :-) ).  For instance, the end of the sentence in the approved version is where you find that the prime in question is a positive whole number, but I prefer the draft version where it comes right out and tells you that.  I think a similar modification to clarify that the whole number is bigger than 1 from the outset, "baldly" saying it, as Noel suggested, is also in order.  I also like your suggestion, Noel, of providing clearer reasons for 1's unique position.  Would that be better as a new subsection, a footnote, or a link to a page about the arithmetic properties of 1?[[User:Barry R. Smith|Barry R. Smith]] 22:31, 31 March 2008 (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 <math>N>1</math>, 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. --[[User:Catherine Woodgold|Catherine Woodgold]] 07:30, 9 May 2007 (CDT)
:::: I'd say a new subsection, not a footnote. Although I don't know where it would fit... hmmmmm (cogitates). Maybe take the third para of the intro, about factorization, and move it to a new section immediately after the intro, titled something like "Factorization and primes"; I think that's a sufficiently important aspect of primes that it's worth of a section on its own. Mention of the Fundamental Theorem of Arithmetic would go there, after which it would be natural to flow from that into your point about the FToA ruling out 1 as a prime. The existing text about "(although this is a matter of [the] definition [of a prime], and mathematicians in the past often did consider 1 to be a prime)" would naturally fit in there too. In fact, maybe a sub-section of that "Factorization and primes" section would cover the primality of 1, and although it would start with the FToA point, etc, you could add your other points above about algebraic number theory, etc. [[User:J. Noel Chiappa|J. Noel Chiappa]] 00:38, 1 April 2008 (CDT)


::Good point that we have to add "greater than 1".
::::: Yes, counting is hard ;) I added "greater than 1" to the first sentence, so that's settled for now.
::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 <math>N</math> has a smallest divisor greater than 1, call it <math>q</math>, 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.
::::: Noel's suggestion to have a new section on the Fundamental Theorem of Arithmetic looks like a good idea. We probably don't want to write too much on it, I think details should go at [[unique prime factorization]] or some other article, but I agree that it's important enough in this context to get a section. Indeed, the primality of 1 can covered there, though I'm not sure it should be a sub-section; how much should we say about it? -- [[User:Jitse Niesen|Jitse Niesen]] 08:10, 1 April 2008 (CDT)
::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. -- [[User:Jitse Niesen|Jitse Niesen]] 10:19, 9 May 2007 (CDT)


== Historical remark regarding 1 ==
:::::: Since I'm not a mathematician, and the article is intended (mostly!) for non-mathematicians, would you like me to try the layout I suggested; you all can then check it to make sure I didn't commit any math howlers? [[User:J. Noel Chiappa|J. Noel Chiappa]] 10:32, 1 April 2008 (CDT)


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. [[User:Greg Woodhouse|Greg Woodhouse]] 11:55, 7 May 2007 (CDT)
::::::: Sounds good to me Noel [[User:Barry R. Smith|Barry R. Smith]] 11:40, 1 April 2008 (CDT)


:MathWorld [http://mathworld.wolfram.com/PrimeNumber.html 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)". [[User:Fredrik Johansson|Fredrik Johansson]] 12:00, 7 May 2007 (CDT)
OK, I've taken a crack at it. I hope you will all find the result (mostly :-) satisfactory; it seems to me (at least :-) to flow well, and in a natural progression. A couple of things where I don't have enough math knowledge to really fill in, and you all need to backstop: i) explain some about ''why'' and ''how'' the FToA is so important, ii) some of the more advanced stuff about why 1 is not a prime (in Barry's original comments in this section above) was way over my head, so I just cut-n-pasted the brief allusion here, which you all ought to expand a teensy bit (and make sure my copyediting didn't produce bogosities). Oh, also, the section on factorization should include a sentence or two about how factorization of very large numbers is a key in the crypto-system stuff we alluded to in the intro. I'm too lazy to do that - off to other things! [[User:J. Noel Chiappa|J. Noel Chiappa]] 12:53, 1 April 2008 (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. [[User:Greg Woodhouse|Greg Woodhouse]] 19:20, 7 May 2007 (CDT)
: I think it looks great, Noel. The only concern I have is the statement that the Fundamental Theorem of Arithmetic is an important building block in many areas of number theory. Historically, the Fundamental Theorem appeared in Euclid's "Elements", the most influential math book of all time, as Proposition 14 in Book IX (This is from a secondary source).  Actually, this proposition only shows that if a number n factors as n = p_1 x p_2 x p_3 x ... x p_r, where p_1, ..., p_r are DISTINCT prime numbers (i.e., n could be 30 = 2 x 3 x 5, but not 12 = 2 x 2 x 3, since 2 appears twice), then then those are the only prime numbers that appear in its factorization.  Thus, this says significantly less than the Fundamental Theorem of Arithmetic, and only says something about very special types of numbers.
:It wasn't until about 2000 years after Euclid that the Fundamental Theorem was codified and decisively proved, by Carl Friedrich Gauss (I have seen this claim many times, but don't have a math historian to use as a source). It seems generally believed that earlier people understood the principle of unique factorization, but perhaps there had never been a reason to try to prove it.  It wasn't until larger number systems than the integers began to be considered that it was realized that the Fundamental Theorem describes a particular property of the integers.  In fact, in other number systems, the analog of unique factorization FAILS to be true, which is what Gauss realized and motivated him to prove the theorem for integers. So in a sense, it is the failure of the Fundamental Theorem to be an important result in these other number systems (i.e., it's just not true) that prompted its formulation.
:  Does this make sense?  If so, then maybe I will just stick a brief mention of some of this information in place of the statement that I objected to.  In any case, besides being an assumed property of the integers that is used to build up many of the important results in Arithmetic, I suppose an answer to your question of why FToA is important is that it fails in other number systems. In response to your other question, I don't see any "bogosities" :). <small>...said</small> [[User:Barry R. Smith|Barry R. Smith]] ([[User_talk:Barry R. Smith|talk]]) {{#if:17:42, 1 April 2008|17:42, 1 April 2008|}} (<small>''Please sign your talk page posts by simply adding four tildes, ''</small><nowiki>~~~~</nowiki>.)


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? [[User:Michael Hardy|Michael Hardy]] 18:56, 7 May 2007 (CDT)
:: Got it. My text about ''the Fundamental Theorem of Arithmetic, which is a key building block in many important areas of number theory'' was in large part a reaction to the very name - I figured anything called the Fundamental Theorem of Arithmetic had to be important! But I notice you say "besides being an assumed property of the integers that is used to build up many of the important results in Arithmetic", so perhaps I wasn't so far wrong? :-)
:: So, I'll change the text to say "Fundamental Theorem of Arithmetic, which is used to build up many of the important results in the area of arithmetic", and you can further tweak that to your satisfaction, to be perfectly accurate.
:: After thinking about it, I would suggest that this article probably isn't the place to mention how the FToA is not true in other number systems, because it's one further step removed from the article's focus, which is primes. It would also intrude into the flow from i) the mention of FToA to ii) how the FToA makes it desirable to exclude 1 from the set of primes. That observation would of course be a perfect fit in the [[Fundamental Theorem of Arithmetic]] article, though.
:: I'll also add that remark about how factorization is what's important in public-key crypto work. And then I leave it to you all... :-) [[User:J. Noel Chiappa|J. Noel Chiappa]] 19:28, 1 April 2008 (CDT)


:I have no clue. As I said before, the history of mathematics isn't my strong point (or even close, really). [[User:Greg Woodhouse|Greg Woodhouse]] 19:20, 7 May 2007 (CDT)
:::I do have some remarks and questions.
:::# You added the word "different" to the first sentence, so that it reads: "A prime number is a whole number greater than 1 that can be evenly divided by only two different positive whole numbers, namely 1 and itself." Is this necessary? Is there really a chance that if we remove the word "different", somebody will think that 7 is not a prime because it has three divisors, namely 1, 7 and 1?
:::# We have an article about [[unique factorization]]. Do we need a different article about the [[Fundamental Theorem of Arithmetic]]?
:::# You say that the Fundamental Theorem of Arithmetic "is used to build up many of the important results in the area of arithmetic." I would replace "arithmetic" by "number theory". Arithmetic can mean number theory, but I think this meaning is disappearing. Barry, would do you think?
:::# Finally, I have my doubts about "a more general trend in mathematics over the past century, which is to recognize that 0 and 1 are very special numbers". I thought mathematicians always recognized this. I think the point in the posts you refer to is that mathematicians are getting more careful to make sure that their proofs are really correct.
:::I'd write longer paragraphs, but that's personal I guess. I have a maths history book which says that Gauss proved the Fundamental Theorem. I think earlier proofs exist, but they are nowadays deemed incomplete. I didn't know that the Elements contains a more restricted result. -- [[User:Jitse Niesen|Jitse Niesen]] 14:18, 3 April 2008 (CDT)


== Huh? ==
:::: Reponses:
::::# I agree, the word different is superfluous.
::::# No, we don't need two separate articles.  However, if the main article is going to be "unique factorization", then "FToA" should be given as a synonym in the first sentence.  Furthermore, typing "FToA" into CZ's main search box should deposit one on that page.  Right now, if one searches for "FToA", he just gets a list of search hits, and the first one is actually the prime number page draft, not the unique factorization page.  It needs to be redirected -- is this easy? I'll look into it... (Interestingly, typing "unique factorization" into the Wikipedia search box deposits you on a page about unique factorization domains -- a bad choice of redirection IMO).
::::#  Noel originally wrote "number theory", and my long winded response boiled down to my thinking "arithmetic" was the more apt word.  Perhaps it should say "elementary number theory".  The problem is that number theory these days is big, and for instance, it seems hard to me to draw a direct connection between unique factorization and major results in analytic number theory (although Euler's factorization of the zeta function involves it).  Also, it is precisely the failure of unique factorization that spurred the invention of rings and ideals and algebraic number theory in general (although unique factorization into prime IDEALS is an important building block in this area).  Certainly, if you stick to elementary number theory, working with integers and congruences, then it is important, although even here it is hard to gauge how much.  Very few proofs seem to come out and say, "and this next step follows from unique factorization".  It is more "fundamental" in its importance, since much structure would be absent if it were false.  For instance, statements that "such and such type of number has a prime factor of this type" would be silly, if you could have different prime factorizations.  Also, security of certain cryptosystems is based on the assumption that the product of two large primes is hard to factor.  If there were other factorizations, some that maybe involved small primes, these would no longer be viable cryptosystems.  There must be a pithy way to summarize this type of fundamental importance accurately, and if someone thinks of one, that would be great.  Maybe a nice allusion to that atoms/molecules metaphor again?  I'll try to figure one out myself.
::::# I agree that even mathematical noobs probably have always realized that 0 and 1 are "special" numbers right away.  However, even important number theorists as late as the 1900's would sometimes list 1 as a prime number.  No one found a real need for a proof of FToA until Gauss, so I wouldn't say necessarily that the problem was that proofs were incorrect.  I would guess that once Gauss proved this result, he did not consider 1 to be prime (I'd hope so, at least).    Perhaps a lot of the time, people didn't realize the appropriate definitions until enough of the theory had been developed, like the general theory of rings and the FToA in this case  ("appropriate" definitions being ones that make the theory and theorems as simple and elegant as possible).
::::I get worried about repeating something like "Gauss was the first to prove this theorem".  I have been tempted to write sentences like this several times now, and even if I see one in a history book, there never seems to be a source.  How could there be?  Someone would have had to comb every extant reference to make sure no one else had proved it earlier, an impossible task.  So every statement like this seems to beg the question, "how many old primary sources did you consult before you decided that Gauss was the first?"  It certainly seems to happen regularly that some new source is discovered that proves such a statement false.[[User:Barry R. Smith|Barry R. Smith]] 22:04, 3 April 2008 (CDT)


This has been added to the "draft":
:::: A few addtional bits:
::::# I added that "different" after thinking for a while of how to make the intro sentence as clear ''and'' consise as possible for ''non-mathematician'' readers; i.e. I only added words where I thought they really helped. Yes, ''technically'' it's superfluous (to the likes of us :-) - but I believe it will increase the likelihood of correct comprehension for the 'average' reader.
::::# For what it's worth, Wikipedia has a separate article on the {{WP|Fundamental Theory of Mathematics}}. Not saying we need one, that's y'all's call, just providing data. I have set up the redirects as you suggest.
::::# I believe I can find a way to say exactly what Barry wants - that although the FToA is not called out specifically, what it says is important, and the basic idea/attribute of the integers (unique factorizability) which it talks about is used throughout "elementary number theory" (which I will also put in).
::::# That was my (not very good, sigh) attempt to capture the essence of what Conway said in one of his posts in that thread ("Mathematicians this century are generally much more careful about exceptional behavior of numbers like 0 and 1 than were their predecessors: we nowadays take care to adjust our statements so that our theorems are actually true. It's easy to find lots of statements in 19th century books that are actually false with the definitions their authors used"). I tried to capture the jist of what he said in a concise way that would make sense to non-experts, although you're right, my words went further than what he said - I will look at it again.
:::: Thanks for being so patient with this non-mathematician! :-) [[User:J. Noel Chiappa|J. Noel Chiappa]] 23:02, 3 April 2008 (CDT)


:: ...
:::: OK, fixed #'s 3 and 4. See what you think. [[User:J. Noel Chiappa|J. Noel Chiappa]] 23:20, 3 April 2008 (CDT)
:: 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. [[User:Michael Hardy|Michael Hardy]] 15:13, 8 May 2007 (CDT)
::::: Yes, perhaps I am not well-suited to decide how useful the word "different" is.  If it makes it clearer to the non-expert, like explicitly clarifying that 1 is not prime, then perhaps it is better to include it.  I like having the main page titled "unique factorization" rather than FToA, since the former is the more descriptive term.  I definitely like the change to "elementary number theory".  However, that sentence now seems wordy to me.  Do you like this  better: "Unique factorization into prime numbers is formalized as the Fundamental Theorem of Arithmetic, being the foundation for much of the structure of the integers described by elementary number theory."?  Also, I will go ahead and divide the first paragraph of the section on 1 into two sentences -- it seems too long for one sentence.  I understand now where your comment on 0 and 1 came from -- I had already forgotten precisely what Conway had written (absent-minded professorism!).  I like that sentence now. Can I cut the last statement in that section now? It can appear on some more advanced page, and the words "the Fundamental Theorem is a good example" above it seem to obviate the need for having this second more obscure example. 
::::: Also, thanks for wading through my windiness to help make this fundamental page correct but accessible to a broad readership.[[User:Barry R. Smith|Barry R. Smith]] 10:58, 4 April 2008 (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! [[User:Greg Woodhouse|Greg Woodhouse]] 17:20, 8 May 2007 (CDT)
:::::: Hi, sorry about the slow reply - been on travel.
:::::: I will take a look at that sentence; I already trimmed it once to be less wordy.
:::::: The two-sentence form looks better to me.
:::::: The "last statment in that section" (i.e. "At a more advanced level ... not visible in the context of integers.") could definitely move to a more advaced page.
:::::: Hey, we're all here to make a better encyclopaedia, right? :-) [[User:J. Noel Chiappa|J. Noel Chiappa]] 16:19, 10 April 2008 (CDT)


== Scriptstyle? ==
:::::: OK, I took a crack at that section. I split that sentence you were asking about into two, but made no other changes to it because it didn't seem that bad. I did make a number of other changes to improve the overall readability, though.
:::::: First, I had placed that long sentence at the end of the section, so that it would flow into the opening of the next section (why 1 is not a prime). I decided that the advantages of that flow were not worth the overall non-optimal ordering of content within this section, so I moved it up. When I did that, I could run that 'widow' short para about the atom analogy into it, so that analogy was no longer just hanging about on its own (and I could cut some duplicate verbiage at the same time).
:::::: Finally, I reworded the last sentence in the first para; the old version was crisp and exact 'mathematicsese', but I think my new (slightly longer) version will be more accessible to 'average' readers.
:::::: Anyway, see what you think. [[User:J. Noel Chiappa|J. Noel Chiappa]] 16:48, 10 April 2008 (CDT)


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. --[[User:Catherine Woodgold|Catherine Woodgold]] 20:24, 8 May 2007 (CDT)
: ''Reset indentation''
: I like your changes, with two comments:  your rewording of the last sentence in the first para of the approved version is fine, except that some mathematicians encourage avoidance of the word "any", as it can mean different things to different people and in different contexts.  When someone says, "if x is any number", do they mean that they are making a statement that is true for ALL possible values of x, or do they mean that they have singled out ONE particular value of x, whose identity has not yet been revealed, and are making a statement about that one value?  I don't know if this type of confusion can arise in your new sentence.  Perhaps a statement about use of the word "any" should be on the style guidelines for the math workgroup, if it isn't there already? The second comment is that to me, at least, the sentence suggesting the analogy to atoms/molecules seems to show up abruptly at the end of the paragraph, with not enough connection to the earlier sentences in the paragraph.  Good work though.[[User:Barry R. Smith|Barry R. Smith]] 23:54, 18 May 2008 (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 <nowiki><imath>...</imath></nowiki> 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 <nowiki>\scriptstyle</nowiki> - 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 <nowiki>\scriptstyle</nowiki> 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<nowiki> \scriptstyle</nowiki> ''over and over'', when someone else likely to come along and take it all out isn't exactly fun. [[User:Greg Woodhouse|Greg Woodhouse]] 20:45, 8 May 2007 (CDT)
:: I get your point about 'any': I had thought that the wording there ("Every number N >1 ... for any particular N") made it clear that if referred, one at a time, to any (all) N in that set. The problem is that I want the reader to, in their mind, single out one particular N, so I can make a statement about the set of factorizations of ''that'' N (i.e. make it is simple as possible to understand) - and also indicate that that statement is true of all N in the set N > 1. What word(s) do mathematicians use when they want the 'all' meaning of "any"? Or should I reorder it (which will make it a tiny bit longer), and explicitly say something like 'Considering any N > 1, it can be written as a product of prime factors, and all such expressions for N will contain the same factors, differing only in the sequence in which they are listed; this is true for all N > 1.'? That's probably slightly more rigorous, but somehow seems to not flow as well. Or maybe I'm being too picky? Is there some other alternative fix (an added modifier, or something) you can suggest, that removes the possible lack of rigour, while retaining the current form?
:: As to the analogy to atoms/molecules (and associated footnote), I agree with you! I didn't introduce any of that; I wasn't too thrilled by it, but out of deference to whoever wrote it, didn't want to simply toss it. I struggled to find a place for it where it ''did'' fit, and following onto the text about "critical building block in many of the important results in the area of elementary number theory" seemed the best I could find. One possibility is to transplant it up into the intro section, placing it in a para of its own, after the para about odd numbers, where I think it would fit a bit better (in terms of overall flow). That would make an awfully short one-sentence para, though. I suppose we could move some (or all?) of the text in the footnote up there, to bulk it up, but if so, I should probably rewrite the whole thing from scratch, to come up to the standards we have reached everywhere else (preen, preen :-). Let me ask this: is the thought it is trying to express accurate, and worth retaining? If so, I can set to and try and write it better, and would suggest that location in the intro for it. Reaction? [[User:J. Noel Chiappa|J. Noel Chiappa]] 10:25, 19 May 2008 (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.
==Primes of special forms subsection==


I am in favor of using the best tool for the jobIf that means using tools in ways that aren't their intended purpose, big deal.  Sorry, call me a philistineI never cared about how H1-H6 were ''intended'' to be used in HTML, either.  I ''greatly'' prefer the aesthetics of the use of <nowiki>\scriptstyle</nowiki> in my browsers of choiceIf 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.
The third type of prime considered in this section seems out of place to meAs far as I know, primes of the form n^2+1 are mostly a curiosity, and uninteresting for anything elseIt would be easy to find many other types of "primes" of this formAny one object to me removing them?  As consolation, I am going to insert a bullet about primes in arithmetic sequences, which seem much more important to me.[[User:Barry R. Smith|Barry R. Smith]] 17:50, 1 April 2008 (CDT)


This is just my own nonbinding opinion, and I think the decision ought to be left up to the mathematics workgroup. --[[User:Larry Sanger|Larry Sanger]] 20:54, 8 May 2007 (CDT)
: <s>I might suggest leaving them because Mersenne primes are a major source of "largest prime" values. I know, I know, to most true mathematicians this is probably boring juvenile crap, but some people enjoy it (like some people enjoy tiddliwinks) - see, for instance, [http://www.mersenne.org GIMP] so an article on primes ought to at least ''mention'' Mersenne primes!</s> Blast, wrote that whole section, then realized you were talking about ''N^2+1'', not ''2^N-1''! On thinking about it, they seem rather obscure to me, so we can probably lose them. I'd just comment them out, rather than discard the text completely; that way, if someone makes a case for them, it's trivial to put them back. [[User:J. Noel Chiappa|J. Noel Chiappa]] 19:40, 1 April 2008 (CDT)


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


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? --[[User:Larry Sanger|Larry Sanger]] 21:04, 8 May 2007 (CDT)
I am interested in having a non-mathematician perspective on the last part of this section.  It seems to me to ramp up in sophistication very quickly, starting with mention of the word "ideal", and then moving into sentences about "rings" and "generation" of "ideals".  Certainly, if this page is intended for non-specialists, then those terms should at least be linked.  But would they be better placed in a page about prime ideals in rings, and a much simplified discussion put in its place on this page?[[User:Barry R. Smith|Barry R. Smith]] 17:50, 1 April 2008 (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. [[User:Michael Hardy|Michael Hardy]] 13:26, 9 May 2007 (CDT)
: I couldn't make heads or tails of it most of it. I would just move everything past the Euclid's Lemma thing to an /Advanced subpage (see [[Quantum mechanics/Advanced]] for what such pages look like). The first couple of paras I would leave, as they are comprehensible to us mortals, and mildly interesting - but I will rewrite them a bit to make them clearer. [[User:J. Noel Chiappa|J. Noel Chiappa]] 19:48, 1 April 2008 (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 [[CZ:Formatting_mathematics/Theorem_capitalization|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. - [[User:Greg Martin|Greg Martin]] 16:26, 10 May 2007 (CDT)
==Fermat primes==
The current discussion of Fermat primes says that one can construct a regular p-gon if p is a Fermat prime. Perhaps more surprising is that these are the ONLY primes for which you can construct a regular p-gon, so I am going to add this.


==APPROVED Version 1.1==
The above unsigned comment isn't by me (though I agree with it). [[User:Peter Jackson|Peter Jackson]] 17:12, 24 November 2008 (UTC)
Congratulations on V1.1!  --[[User:D. Matt Innis|Matt Innis]] [[User talk:D. Matt Innis|(Talk)]] 19:35, 10 May 2007 (CDT)
<div class="usermessage plainlinks">Discussion for [http://en.citizendium.org/wiki?title=Prime_number/Draft&direction=next&oldid=100099364 Version 1.1] stopped here. Please continue further discussion under this break. </div>


== First paragraph ==
MUCH MORE IMPORTANT. There's an error in the article. It says Fermat primes are 2↑n + 1. That should be 2↑2↑n + 1. [[User:Peter Jackson|Peter Jackson]] 17:10, 24 November 2008 (UTC)


It says, ''" In particular, a prime number p cannot be factored as the product of two numbers <math>\scriptstyle p=a\times b</math>"''.  This is not true in general. It should say "two whole numbers" or "two integers" rather than "two numbers". --[[User:Catherine Woodgold|Catherine Woodgold]] 07:43, 11 May 2007 (CDT)
==Other types of primes==
I think that if we are going to discuss other types of primes, obvious choices include Wieferich and Wilson primes.  But where do we draw a line about which special types of primes to include?  Wieferich primes showed up in work on Fermat's Last TheoremPerhaps a criterion would be to include any special forms for which a significant result is known? I like this better than the criterion of including any forms of primes with "names".[[User:Barry R. Smith|Barry R. Smith]] 18:15, 1 April 2008 (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.
: Write articles on them, and link to those articles from the "Related Articles" subpage. As you suggest, I'd only put really important ones here. That's probably where to put Fermat primes, now that I think of it. I'll leave that to you all too. [[User:J. Noel Chiappa|J. Noel Chiappa]] 19:48, 1 April 2008 (CDT)


: 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.) - [[User:Greg Martin|Greg Martin]] 11:55, 11 May 2007 (CDT)
== Reapproval needed ==


::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 whichWell, 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 <math> 5 = -5 \times -1.</math>, which would be OK I suppose if primes were defined differently. --[[User:Catherine Woodgold|Catherine Woodgold]] 18:33, 11 May 2007 (CDT)
As pointed about by Peter Jackson above, the definition of Fermat prime should be a prime of the form <math>2^{2^n}+1</math>, not one of the form <math>2^n+1</math>.  Actually, the primes of the first type ''are precisely'' the primes of the second type. (Proof: If n=kt with k an odd prime, then  
2^n+1 has 2^t+1 as a factor other than 1 and itself)But as written, the definition is certainly nonstandard and somewhat misleading. As such, I think reapproval of this page should be expedited.


::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."''
Issues:   


::In the third sentence:  ''"In particular"'' seems out of context to me.  I would just delete it.  --[[User:Catherine Woodgold|Catherine Woodgold]] 08:23, 12 May 2007 (CDT)
* Noel suggests that Fermat primes should appear just on the "related articles" page.  I think it is appropriate to have something in the main article about special types of primes, mentioning a few typesFermat prime is certainly one of the most popular.  But there should be agreement -- the problem could be fixed just by linking to pages about a few types of special primes without defining any of them in the "prime number" page.


:::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.
* It is technically correct as written, so if reapproval is an arduous process (I haven't done it myself), then perhaps it is not worth it just to fix this one problem.
:::I deleted "in particular" as suggested. -- [[User:Jitse Niesen|Jitse Niesen]] 04:59, 16 May 2007 (CDT)


== Proof by contradiction ==
* The current draft is rather different than the approved version, so if there is agreement that the draft is better than the current version, or could be made better than it, then perhaps we should work to reapprove now even if the process is difficult.[[User:Barry R. Smith|Barry R. Smith]] 21:22, 27 November 2008 (UTC)
 
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.''
 
--[[User:Catherine Woodgold|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&nbsp;+&nbsp;the product of the first several primes is necessarily prime.  Note that 2&times;3&times;5&times;7&times;11&times;13&nbsp;+&nbsp;1 is composite, since it is 59&times;509.  If Euclid weren't such a big name, maybe "Proof number 1" would be seen more often. [[User:Michael Hardy|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". -- [[User:Jitse Niesen|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 "<math>1 = 0</math>", 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 <math>\scriptstyle \{ p_1, p_2, p_3, \ldots, p_n \}</math>.''
 
:::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 <math>N + 1</math> is prime, though.  That misunderstanding happens (or doesn't) in the middle of the proof, which can be the same in both cases.  --[[User:Catherine Woodgold|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. -- [[User:Jitse Niesen|Jitse Niesen]] 08:05, 12 May 2007 (CDT)
 
:::::Thanks.  Your "This shows that..." works better then my "Therefore...".  --[[User:Catherine Woodgold|Catherine Woodgold]] 08:31, 12 May 2007 (CDT)
 
=== Book IX, Proposition 20 ===
 
At [http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html 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. [[User:Michael Hardy|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. :-) [[User:Greg Woodhouse|Greg Woodhouse]] 19:50, 12 May 2007 (CDT)

Latest revision as of 16:22, 27 November 2008

This article has a Citable Version.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
To learn how to update the categories for this article, see here. To update categories, edit the metadata template.
 Definition A number that can be evenly divided by exactly two positive whole numbers, namely one and itself. [d] [e]
Checklist and Archives
 Workgroup category Mathematics [Please add or review categories]
 Talk Archive 1, 2  English language variant British English

What kind of number

The first sentence doesn't specify what kind of numbers we are dealing with. Andres Luure 22:26, 5 November 2007 (CST)

It says "A prime number is a number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself." The word "itself" implies it must be a positive whole number. (But maybe it could be more explicit.) Michael Hardy 14:03, 20 December 2007 (CST)
And why not be more explicit? It seems much clearer to me, a number theorist even, if you say "a prime number is a positive whole number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself". In the original version, I didn't immediately return to the beginning of the sentence and think, "oh, that IMPLIES that the original number was a positive whole number". If the only concern is that repeating the same phrase twice might be a turn off, let me quote from Strunk and White's "Elements of Style", one of the guides to style that we are supposed to take advice from (see section 19) "The likeness of form enables the reader to recognize more readily the likeneses of content and function...the unskilled writer often violates this principle mistakenly believing in the value of constantly varying the form of expression. When repeating a statement to emphasize it, the writer may need to vary its form. Otherwise, the writer should follow the principle of parallel construction." It seems to me that the repetition in our case is not for emphasis, and after the repetition, it will be firmly fixed in the reader's mind that all numbers begin considered are positive whole numbers.Barry R. Smith 20:40, 29 March 2008 (CDT)

I am not sure it is even clear to say "exactly two" when it could be "two and only two." --Thomas Simmons 19:56, 8 November 2007 (CST)

In mathematics, at least, the phrases "there are exactly two" and "there are precisely two" are understood to express the same statement as "there are two, and only two" (for instance, see the discrete math text I taught out of this past term, or the Wiki page on if and only if). I have considered these as equivalent for many years, so it is hard for me to put myself in the shoes of someone who might be seeing this for the first time. The issue that you are concerned with is that someone might accidentally confuse "exactly two" with the idea that it has at least two, but possibly more, positive divisors. I cannot see how even people with very little mathematical experience would interpret "exactly two" in this manner, the word "exactly" being inserted exactly (hehe) to let you know that this is the precise number. Furthermore, I think it is hard to argue against "exactly two" being the more elegant phrase. I much prefer the phrasing of the first sentence of the approved article to the first sentence of the current draft. Any rebuttals?Barry R. Smith 20:40, 29 March 2008 (CDT)
I agree. Even to naive readers "exactly two" cannot possibly mean "at least two". J. Noel Chiappa 22:12, 29 March 2008 (CDT)

Footnotes versus links to stubs: Until many of the terms are explained with their own articles, the use of footnotes to explain terms and analogies should continue. Otherwise we will have dead links in red letters for a long time to come. This will also mean that the article can reach a broader population as it is written. The chemistry metaphor is another example, that comparisons might be lost on anyone who is not up to speed on freshman level chemistry. So for ease of use and market appeal and just plain educational focus, explaining terms in footnotes would be a good idea. The high school students using CZ today will be the grad students referring to it in future.--Thomas Simmons 19:56, 8 November 2007 (CST)

What i miss

There are some things, that are not in thearticle:

--arbol01 05:04, 1 January 2008 (CST)

I don't understand the first comment under "Properties of prime numbers"
As for prime divisors of elements of those two particular sequences, it seems to me that these are far too specialized to be included in this page, and would be better placed on the "Perrin sequence" and "Lucas sequence" pages separately. Otherwise, one would need to enumerate ALL named recursively defined sequences, and the divisibility properties in each case. I would imagine that just this task would encompass many pages in itself.Barry R. Smith 00:20, 30 March 2008 (CDT)
    • p is a Prime number <=> p|(p over n) for 1<n<p
Iff p is a prime number, than p divides (p over n) [Binomialcoeffizient] for every n between 1 and p.
If p is a prime number, than p divides Pp and p divides U(P,Q)p - P
are properties, that belongs to the Prime numbers. It belongs to the Perrin sequence respectively to the Lucas requence respectively to the binomialcoefficence too. --Karsten Meyer 02:45, 19 May 2008 (CDT)

1 revisited

Regarding my above comment in "What kind of number" above, I personally feel that the first sentence should read something like, "A prime number is a whole number greater than 1 that can be evenly divided by exactly two positive whole numbers, namely 1 and itself". It seems that the main argument above against saying a prime must be bigger than 1 from the outset is the need for clarity in the first sentence, but I feel that currently this clarity comes at the price of correctness. As written, I feel the that first sentence is plain wrong, and I personally wouldn't put my stamp of approval on it.

I don't want to sound TOO dismissive. I didn't just go edit the draft, because I understand that their was some discussion about this above. Apparently, the status of 1 seems to have been problematic even when the fundamental of arithmetic were laid down in Euclid's "Elements". However, it seems to me that the tone of the approved version suggests that the typical modern "choice" to label 1 as neither prime nor composite is a result of whimsy or chance. This is a false impression.

In a sense, I guess, defining 1 as special can seem as arbitrary as defining 0 factorial to be 1. But with the invention of the gamma function and the recognition of its canonical properties, can there be any dispute as to the correct definition of 0 factorial? Similarly, there are very sound reasons that 1 has been given special status over the last century or so. The easiest to explain is that the Fundamental Theorem of Arithmetic is just false if 1 is considered prime: considere, 6 = 2*3 = 2*3*1 -- two different prime factorizations. (By the way, I also think that the words "Fundamental Theorem of Arithmetic" should appear somewhere on the "prime number" page -- can't remember if I saw it anywhere). A second reason is that with the development of algebraic number theory, the units in algebraic number fields were found to play a very special and important role. Within the integers, 1 and -1 are the only units, so it is hard to get a feel for the special role they play only within this context. Nevertheless, the fact that 1 is the unique multiplicative identity within the integers should make a strong impression. (For more about 1, see this website http://mathforum.org/kb/message.jspa?messageID=1379707, and especially the comments by John Conway, a world-renown number theorist.)

In summary, although the status of 1 might have fluctuated in the past, I believe the consensus of the vast majority of working mathematicians at present is that it should not be considered prime, and this is reflected in todays high-school textbooks. Furthermore, I do not see any indication that this will change soon. Thus, it seems that the proper definition should make it clear that 1 is not prime from the first sentence. Otherwise, we will be spreading disinformation to those casual learners who wonder, "hmm, I wonder if 1 is a prime", look at the first line of the Citizendium page, and then wander off to tell their friends what they learned.Barry R. Smith 01:31, 30 March 2008 (CDT)

Dude, you're the expert! I (at least, can't speak for everyone) defer to your clear familiarity. So I'd go for it. Plus to which, your point about the Fundamental Theorem of Arithmetic is good (and so easily understandably by all that it should probably be mentioned in the article as a reason why 1 is not considered by mathematicians as being part of the set of prime numbers, even though by the simplistic definition of 'prime', it seems to be prime). J. Noel Chiappa 11:40, 30 March 2008 (CDT)
Barry, the first sentence currently says "A prime number is a positive whole number that can be evenly divided by exactly two positive whole numbers, namely 1 and itself." I believe this does say that 1 is not prime, just as you want, as the number 1 has only one divisor, namely 1 itself. So I'm not sure what your point is.
I agree with all the rest you wrote (for what it's worth, as you know of course more number theory than I do). -- Jitse Niesen 16:00, 30 March 2008 (CDT)
Someone careful and analytic might draw that conclusion, but not all our readers might fit that definition. Baldly saying '1 is not a prime number' is probably what they need. Without in any way intended to be demeaning to them, I am always mindful of that wonderful George Carlin line: "Think of how dumb the average person is - and then realize that half of them are dumber than that."
(Adding "different" - as in "two different positive whole numbers" - might make the definition cast-iron, though). But it might still be useful to have a section on 'why 1 is not a prime number'; the point about the Fundamental Theorem of Arithmetic could go there. J. Noel Chiappa 23:31, 30 March 2008 (CDT)
Yes, Jitse, it seems that after all of that, I understand 1 okay but I still have trouble counting to 2 :-). Anyway, my own error emphasizes the point that inferring information about the prime number from information presented at the END of the sentence is not my own thought process, and probably not a lot of other people's. (I suppose if I still kept up my German, I would be used to that sort of thing :-) ). For instance, the end of the sentence in the approved version is where you find that the prime in question is a positive whole number, but I prefer the draft version where it comes right out and tells you that. I think a similar modification to clarify that the whole number is bigger than 1 from the outset, "baldly" saying it, as Noel suggested, is also in order. I also like your suggestion, Noel, of providing clearer reasons for 1's unique position. Would that be better as a new subsection, a footnote, or a link to a page about the arithmetic properties of 1?Barry R. Smith 22:31, 31 March 2008 (CDT)
I'd say a new subsection, not a footnote. Although I don't know where it would fit... hmmmmm (cogitates). Maybe take the third para of the intro, about factorization, and move it to a new section immediately after the intro, titled something like "Factorization and primes"; I think that's a sufficiently important aspect of primes that it's worth of a section on its own. Mention of the Fundamental Theorem of Arithmetic would go there, after which it would be natural to flow from that into your point about the FToA ruling out 1 as a prime. The existing text about "(although this is a matter of [the] definition [of a prime], and mathematicians in the past often did consider 1 to be a prime)" would naturally fit in there too. In fact, maybe a sub-section of that "Factorization and primes" section would cover the primality of 1, and although it would start with the FToA point, etc, you could add your other points above about algebraic number theory, etc. J. Noel Chiappa 00:38, 1 April 2008 (CDT)
Yes, counting is hard ;) I added "greater than 1" to the first sentence, so that's settled for now.
Noel's suggestion to have a new section on the Fundamental Theorem of Arithmetic looks like a good idea. We probably don't want to write too much on it, I think details should go at unique prime factorization or some other article, but I agree that it's important enough in this context to get a section. Indeed, the primality of 1 can covered there, though I'm not sure it should be a sub-section; how much should we say about it? -- Jitse Niesen 08:10, 1 April 2008 (CDT)
Since I'm not a mathematician, and the article is intended (mostly!) for non-mathematicians, would you like me to try the layout I suggested; you all can then check it to make sure I didn't commit any math howlers? J. Noel Chiappa 10:32, 1 April 2008 (CDT)
Sounds good to me Noel Barry R. Smith 11:40, 1 April 2008 (CDT)

OK, I've taken a crack at it. I hope you will all find the result (mostly :-) satisfactory; it seems to me (at least :-) to flow well, and in a natural progression. A couple of things where I don't have enough math knowledge to really fill in, and you all need to backstop: i) explain some about why and how the FToA is so important, ii) some of the more advanced stuff about why 1 is not a prime (in Barry's original comments in this section above) was way over my head, so I just cut-n-pasted the brief allusion here, which you all ought to expand a teensy bit (and make sure my copyediting didn't produce bogosities). Oh, also, the section on factorization should include a sentence or two about how factorization of very large numbers is a key in the crypto-system stuff we alluded to in the intro. I'm too lazy to do that - off to other things! J. Noel Chiappa 12:53, 1 April 2008 (CDT)

I think it looks great, Noel. The only concern I have is the statement that the Fundamental Theorem of Arithmetic is an important building block in many areas of number theory. Historically, the Fundamental Theorem appeared in Euclid's "Elements", the most influential math book of all time, as Proposition 14 in Book IX (This is from a secondary source). Actually, this proposition only shows that if a number n factors as n = p_1 x p_2 x p_3 x ... x p_r, where p_1, ..., p_r are DISTINCT prime numbers (i.e., n could be 30 = 2 x 3 x 5, but not 12 = 2 x 2 x 3, since 2 appears twice), then then those are the only prime numbers that appear in its factorization. Thus, this says significantly less than the Fundamental Theorem of Arithmetic, and only says something about very special types of numbers.
It wasn't until about 2000 years after Euclid that the Fundamental Theorem was codified and decisively proved, by Carl Friedrich Gauss (I have seen this claim many times, but don't have a math historian to use as a source). It seems generally believed that earlier people understood the principle of unique factorization, but perhaps there had never been a reason to try to prove it. It wasn't until larger number systems than the integers began to be considered that it was realized that the Fundamental Theorem describes a particular property of the integers. In fact, in other number systems, the analog of unique factorization FAILS to be true, which is what Gauss realized and motivated him to prove the theorem for integers. So in a sense, it is the failure of the Fundamental Theorem to be an important result in these other number systems (i.e., it's just not true) that prompted its formulation.
Does this make sense? If so, then maybe I will just stick a brief mention of some of this information in place of the statement that I objected to. In any case, besides being an assumed property of the integers that is used to build up many of the important results in Arithmetic, I suppose an answer to your question of why FToA is important is that it fails in other number systems. In response to your other question, I don't see any "bogosities" :). ...said Barry R. Smith (talk) 17:42, 1 April 2008 (Please sign your talk page posts by simply adding four tildes, ~~~~.)
Got it. My text about the Fundamental Theorem of Arithmetic, which is a key building block in many important areas of number theory was in large part a reaction to the very name - I figured anything called the Fundamental Theorem of Arithmetic had to be important! But I notice you say "besides being an assumed property of the integers that is used to build up many of the important results in Arithmetic", so perhaps I wasn't so far wrong? :-)
So, I'll change the text to say "Fundamental Theorem of Arithmetic, which is used to build up many of the important results in the area of arithmetic", and you can further tweak that to your satisfaction, to be perfectly accurate.
After thinking about it, I would suggest that this article probably isn't the place to mention how the FToA is not true in other number systems, because it's one further step removed from the article's focus, which is primes. It would also intrude into the flow from i) the mention of FToA to ii) how the FToA makes it desirable to exclude 1 from the set of primes. That observation would of course be a perfect fit in the Fundamental Theorem of Arithmetic article, though.
I'll also add that remark about how factorization is what's important in public-key crypto work. And then I leave it to you all... :-) J. Noel Chiappa 19:28, 1 April 2008 (CDT)
I do have some remarks and questions.
  1. You added the word "different" to the first sentence, so that it reads: "A prime number is a whole number greater than 1 that can be evenly divided by only two different positive whole numbers, namely 1 and itself." Is this necessary? Is there really a chance that if we remove the word "different", somebody will think that 7 is not a prime because it has three divisors, namely 1, 7 and 1?
  2. We have an article about unique factorization. Do we need a different article about the Fundamental Theorem of Arithmetic?
  3. You say that the Fundamental Theorem of Arithmetic "is used to build up many of the important results in the area of arithmetic." I would replace "arithmetic" by "number theory". Arithmetic can mean number theory, but I think this meaning is disappearing. Barry, would do you think?
  4. Finally, I have my doubts about "a more general trend in mathematics over the past century, which is to recognize that 0 and 1 are very special numbers". I thought mathematicians always recognized this. I think the point in the posts you refer to is that mathematicians are getting more careful to make sure that their proofs are really correct.
I'd write longer paragraphs, but that's personal I guess. I have a maths history book which says that Gauss proved the Fundamental Theorem. I think earlier proofs exist, but they are nowadays deemed incomplete. I didn't know that the Elements contains a more restricted result. -- Jitse Niesen 14:18, 3 April 2008 (CDT)
Reponses:
  1. I agree, the word different is superfluous.
  2. No, we don't need two separate articles. However, if the main article is going to be "unique factorization", then "FToA" should be given as a synonym in the first sentence. Furthermore, typing "FToA" into CZ's main search box should deposit one on that page. Right now, if one searches for "FToA", he just gets a list of search hits, and the first one is actually the prime number page draft, not the unique factorization page. It needs to be redirected -- is this easy? I'll look into it... (Interestingly, typing "unique factorization" into the Wikipedia search box deposits you on a page about unique factorization domains -- a bad choice of redirection IMO).
  3. Noel originally wrote "number theory", and my long winded response boiled down to my thinking "arithmetic" was the more apt word. Perhaps it should say "elementary number theory". The problem is that number theory these days is big, and for instance, it seems hard to me to draw a direct connection between unique factorization and major results in analytic number theory (although Euler's factorization of the zeta function involves it). Also, it is precisely the failure of unique factorization that spurred the invention of rings and ideals and algebraic number theory in general (although unique factorization into prime IDEALS is an important building block in this area). Certainly, if you stick to elementary number theory, working with integers and congruences, then it is important, although even here it is hard to gauge how much. Very few proofs seem to come out and say, "and this next step follows from unique factorization". It is more "fundamental" in its importance, since much structure would be absent if it were false. For instance, statements that "such and such type of number has a prime factor of this type" would be silly, if you could have different prime factorizations. Also, security of certain cryptosystems is based on the assumption that the product of two large primes is hard to factor. If there were other factorizations, some that maybe involved small primes, these would no longer be viable cryptosystems. There must be a pithy way to summarize this type of fundamental importance accurately, and if someone thinks of one, that would be great. Maybe a nice allusion to that atoms/molecules metaphor again? I'll try to figure one out myself.
  4. I agree that even mathematical noobs probably have always realized that 0 and 1 are "special" numbers right away. However, even important number theorists as late as the 1900's would sometimes list 1 as a prime number. No one found a real need for a proof of FToA until Gauss, so I wouldn't say necessarily that the problem was that proofs were incorrect. I would guess that once Gauss proved this result, he did not consider 1 to be prime (I'd hope so, at least). Perhaps a lot of the time, people didn't realize the appropriate definitions until enough of the theory had been developed, like the general theory of rings and the FToA in this case ("appropriate" definitions being ones that make the theory and theorems as simple and elegant as possible).
I get worried about repeating something like "Gauss was the first to prove this theorem". I have been tempted to write sentences like this several times now, and even if I see one in a history book, there never seems to be a source. How could there be? Someone would have had to comb every extant reference to make sure no one else had proved it earlier, an impossible task. So every statement like this seems to beg the question, "how many old primary sources did you consult before you decided that Gauss was the first?" It certainly seems to happen regularly that some new source is discovered that proves such a statement false.Barry R. Smith 22:04, 3 April 2008 (CDT)
A few addtional bits:
  1. I added that "different" after thinking for a while of how to make the intro sentence as clear and consise as possible for non-mathematician readers; i.e. I only added words where I thought they really helped. Yes, technically it's superfluous (to the likes of us :-) - but I believe it will increase the likelihood of correct comprehension for the 'average' reader.
  2. For what it's worth, Wikipedia has a separate article on the Fundamental Theory of Mathematics. Not saying we need one, that's y'all's call, just providing data. I have set up the redirects as you suggest.
  3. I believe I can find a way to say exactly what Barry wants - that although the FToA is not called out specifically, what it says is important, and the basic idea/attribute of the integers (unique factorizability) which it talks about is used throughout "elementary number theory" (which I will also put in).
  4. That was my (not very good, sigh) attempt to capture the essence of what Conway said in one of his posts in that thread ("Mathematicians this century are generally much more careful about exceptional behavior of numbers like 0 and 1 than were their predecessors: we nowadays take care to adjust our statements so that our theorems are actually true. It's easy to find lots of statements in 19th century books that are actually false with the definitions their authors used"). I tried to capture the jist of what he said in a concise way that would make sense to non-experts, although you're right, my words went further than what he said - I will look at it again.
Thanks for being so patient with this non-mathematician! :-) J. Noel Chiappa 23:02, 3 April 2008 (CDT)
OK, fixed #'s 3 and 4. See what you think. J. Noel Chiappa 23:20, 3 April 2008 (CDT)
Yes, perhaps I am not well-suited to decide how useful the word "different" is. If it makes it clearer to the non-expert, like explicitly clarifying that 1 is not prime, then perhaps it is better to include it. I like having the main page titled "unique factorization" rather than FToA, since the former is the more descriptive term. I definitely like the change to "elementary number theory". However, that sentence now seems wordy to me. Do you like this better: "Unique factorization into prime numbers is formalized as the Fundamental Theorem of Arithmetic, being the foundation for much of the structure of the integers described by elementary number theory."? Also, I will go ahead and divide the first paragraph of the section on 1 into two sentences -- it seems too long for one sentence. I understand now where your comment on 0 and 1 came from -- I had already forgotten precisely what Conway had written (absent-minded professorism!). I like that sentence now. Can I cut the last statement in that section now? It can appear on some more advanced page, and the words "the Fundamental Theorem is a good example" above it seem to obviate the need for having this second more obscure example.
Also, thanks for wading through my windiness to help make this fundamental page correct but accessible to a broad readership.Barry R. Smith 10:58, 4 April 2008 (CDT)
Hi, sorry about the slow reply - been on travel.
I will take a look at that sentence; I already trimmed it once to be less wordy.
The two-sentence form looks better to me.
The "last statment in that section" (i.e. "At a more advanced level ... not visible in the context of integers.") could definitely move to a more advaced page.
Hey, we're all here to make a better encyclopaedia, right? :-) J. Noel Chiappa 16:19, 10 April 2008 (CDT)
OK, I took a crack at that section. I split that sentence you were asking about into two, but made no other changes to it because it didn't seem that bad. I did make a number of other changes to improve the overall readability, though.
First, I had placed that long sentence at the end of the section, so that it would flow into the opening of the next section (why 1 is not a prime). I decided that the advantages of that flow were not worth the overall non-optimal ordering of content within this section, so I moved it up. When I did that, I could run that 'widow' short para about the atom analogy into it, so that analogy was no longer just hanging about on its own (and I could cut some duplicate verbiage at the same time).
Finally, I reworded the last sentence in the first para; the old version was crisp and exact 'mathematicsese', but I think my new (slightly longer) version will be more accessible to 'average' readers.
Anyway, see what you think. J. Noel Chiappa 16:48, 10 April 2008 (CDT)
Reset indentation
I like your changes, with two comments: your rewording of the last sentence in the first para of the approved version is fine, except that some mathematicians encourage avoidance of the word "any", as it can mean different things to different people and in different contexts. When someone says, "if x is any number", do they mean that they are making a statement that is true for ALL possible values of x, or do they mean that they have singled out ONE particular value of x, whose identity has not yet been revealed, and are making a statement about that one value? I don't know if this type of confusion can arise in your new sentence. Perhaps a statement about use of the word "any" should be on the style guidelines for the math workgroup, if it isn't there already? The second comment is that to me, at least, the sentence suggesting the analogy to atoms/molecules seems to show up abruptly at the end of the paragraph, with not enough connection to the earlier sentences in the paragraph. Good work though.Barry R. Smith 23:54, 18 May 2008 (CDT)
I get your point about 'any': I had thought that the wording there ("Every number N >1 ... for any particular N") made it clear that if referred, one at a time, to any (all) N in that set. The problem is that I want the reader to, in their mind, single out one particular N, so I can make a statement about the set of factorizations of that N (i.e. make it is simple as possible to understand) - and also indicate that that statement is true of all N in the set N > 1. What word(s) do mathematicians use when they want the 'all' meaning of "any"? Or should I reorder it (which will make it a tiny bit longer), and explicitly say something like 'Considering any N > 1, it can be written as a product of prime factors, and all such expressions for N will contain the same factors, differing only in the sequence in which they are listed; this is true for all N > 1.'? That's probably slightly more rigorous, but somehow seems to not flow as well. Or maybe I'm being too picky? Is there some other alternative fix (an added modifier, or something) you can suggest, that removes the possible lack of rigour, while retaining the current form?
As to the analogy to atoms/molecules (and associated footnote), I agree with you! I didn't introduce any of that; I wasn't too thrilled by it, but out of deference to whoever wrote it, didn't want to simply toss it. I struggled to find a place for it where it did fit, and following onto the text about "critical building block in many of the important results in the area of elementary number theory" seemed the best I could find. One possibility is to transplant it up into the intro section, placing it in a para of its own, after the para about odd numbers, where I think it would fit a bit better (in terms of overall flow). That would make an awfully short one-sentence para, though. I suppose we could move some (or all?) of the text in the footnote up there, to bulk it up, but if so, I should probably rewrite the whole thing from scratch, to come up to the standards we have reached everywhere else (preen, preen :-). Let me ask this: is the thought it is trying to express accurate, and worth retaining? If so, I can set to and try and write it better, and would suggest that location in the intro for it. Reaction? J. Noel Chiappa 10:25, 19 May 2008 (CDT)

Primes of special forms subsection

The third type of prime considered in this section seems out of place to me. As far as I know, primes of the form n^2+1 are mostly a curiosity, and uninteresting for anything else. It would be easy to find many other types of "primes" of this form. Any one object to me removing them? As consolation, I am going to insert a bullet about primes in arithmetic sequences, which seem much more important to me.Barry R. Smith 17:50, 1 April 2008 (CDT)

I might suggest leaving them because Mersenne primes are a major source of "largest prime" values. I know, I know, to most true mathematicians this is probably boring juvenile crap, but some people enjoy it (like some people enjoy tiddliwinks) - see, for instance, GIMP so an article on primes ought to at least mention Mersenne primes! Blast, wrote that whole section, then realized you were talking about N^2+1, not 2^N-1! On thinking about it, they seem rather obscure to me, so we can probably lose them. I'd just comment them out, rather than discard the text completely; that way, if someone makes a case for them, it's trivial to put them back. J. Noel Chiappa 19:40, 1 April 2008 (CDT)

Alternative definition

I am interested in having a non-mathematician perspective on the last part of this section. It seems to me to ramp up in sophistication very quickly, starting with mention of the word "ideal", and then moving into sentences about "rings" and "generation" of "ideals". Certainly, if this page is intended for non-specialists, then those terms should at least be linked. But would they be better placed in a page about prime ideals in rings, and a much simplified discussion put in its place on this page?Barry R. Smith 17:50, 1 April 2008 (CDT)

I couldn't make heads or tails of it most of it. I would just move everything past the Euclid's Lemma thing to an /Advanced subpage (see Quantum mechanics/Advanced for what such pages look like). The first couple of paras I would leave, as they are comprehensible to us mortals, and mildly interesting - but I will rewrite them a bit to make them clearer. J. Noel Chiappa 19:48, 1 April 2008 (CDT)

Fermat primes

The current discussion of Fermat primes says that one can construct a regular p-gon if p is a Fermat prime. Perhaps more surprising is that these are the ONLY primes for which you can construct a regular p-gon, so I am going to add this.

The above unsigned comment isn't by me (though I agree with it). Peter Jackson 17:12, 24 November 2008 (UTC)

MUCH MORE IMPORTANT. There's an error in the article. It says Fermat primes are 2↑n + 1. That should be 2↑2↑n + 1. Peter Jackson 17:10, 24 November 2008 (UTC)

Other types of primes

I think that if we are going to discuss other types of primes, obvious choices include Wieferich and Wilson primes. But where do we draw a line about which special types of primes to include? Wieferich primes showed up in work on Fermat's Last Theorem. Perhaps a criterion would be to include any special forms for which a significant result is known? I like this better than the criterion of including any forms of primes with "names".Barry R. Smith 18:15, 1 April 2008 (CDT)

Write articles on them, and link to those articles from the "Related Articles" subpage. As you suggest, I'd only put really important ones here. That's probably where to put Fermat primes, now that I think of it. I'll leave that to you all too. J. Noel Chiappa 19:48, 1 April 2008 (CDT)

Reapproval needed

As pointed about by Peter Jackson above, the definition of Fermat prime should be a prime of the form , not one of the form . Actually, the primes of the first type are precisely the primes of the second type. (Proof: If n=kt with k an odd prime, then 2^n+1 has 2^t+1 as a factor other than 1 and itself). But as written, the definition is certainly nonstandard and somewhat misleading. As such, I think reapproval of this page should be expedited.

Issues:

  • Noel suggests that Fermat primes should appear just on the "related articles" page. I think it is appropriate to have something in the main article about special types of primes, mentioning a few types. Fermat prime is certainly one of the most popular. But there should be agreement -- the problem could be fixed just by linking to pages about a few types of special primes without defining any of them in the "prime number" page.
  • It is technically correct as written, so if reapproval is an arduous process (I haven't done it myself), then perhaps it is not worth it just to fix this one problem.
  • The current draft is rather different than the approved version, so if there is agreement that the draft is better than the current version, or could be made better than it, then perhaps we should work to reapprove now even if the process is difficult.Barry R. Smith 21:22, 27 November 2008 (UTC)