Talk:Prime number

Deleted

I struck my own comment as it was useless (due to simple sleight of hand of ref) —Preceding unsigned comment added by Billymac00 (talk • contribs)

Finding an intuitive lead defintion of a prime

The two definitions being argued recently about are quite similar and are equivalent. I think that we will have as many opinions about the best way to express it as there are people involved in the debate, so perhaps a discussion is called for.

My own two cents' worth: If we are to have a natural, intuitive lead-in introducing/defining the concept of a prime for the non-mathematician, perhaps we should have something that relates more directly to the concept that a prime is a number that cannot be factored any further. The two definitions being argued about do not do so - to get to this point, this one has to use logic akin to the fundamental theorem of arithmetic. How about:

A prime number is a natural number other than 1 that cannot be expressed as the product of any pair of natural numbers excluding itself. 1 is excluded since allowing any number that has an inverse introduces unwarranted complexities such as non-terminating/non-unique factorisation.

Quondum (talk) 04:45, 23 October 2011 (UTC)

Not at all intuitive. — Arthur Rubin (talk) 08:07, 23 October 2011 (UTC)

Agreed, too complicated and confusing.--♦IanMacM♦ ^{(talk to me)} 08:28, 23 October 2011 (UTC)

Ah, well, I tried. Aside from deleting my second sentence, I have no better suggestion. I'll let others put forward suggestions/arguments (and watch the fur fly - heh-heh). Quondum (talk) 08:42, 23 October 2011 (UTC)

Here's a concise one, taken from http://mathworld.wolfram.com/PrimeNumber.html (with slight paraphrasing):

A prime number is a natural number having exactly one divisor other than 1.

Just keeping the pot boiling... Quondum (talk) 13:28, 23 October 2011 (UTC)

So, if seeking for natural intuitive lead targeted at non-mathematicians, how about, "natural number n is prime if n items can not be divided up into smaller equal-sized groups of more than one item." ? WillNess (talk) 21:02, 23 October 2011 (UTC)

See my comment below. When we're aiming at non-mathematical readers, putting algebraic variables such as n and p into the text is a bad idea. —David Eppstein (talk) 21:05, 23 October 2011 (UTC)

Yes I saw it, but thought that in this case it was intuitive enough nevertheless. How about "a prime number is that amount of items which can't be divided up into smaller equal-sized groups of more than one item"? WillNess (talk) 12:03, 24 October 2011 (UTC)

No, although that covers everything you really have to think about how that works. And a number is not an amount: a prime number is an abstract concept, it may or may not be used for counting things. Better just to write that it's a [natural] number, and that that it's to do with [mathematical] division, two things learned at a very early age.--JohnBlackburne^words_deeds 13:38, 24 October 2011 (UTC)

A good and for everybody immediately understandable definition must directly and explicitly exclude 1 as a prime. Moreover, why is the definition so strained and why has the introduction no stringency?

Rebecca G (talk) 13:57, 29 October 2011 (UTC)

I think your change is an improvement. I cannot authoritatively comment on the stringency/rigour - however, is it appropriate to be speaking of integers rather than natural numbers, as prime elements within the set of integers is something different? Quondum (talk) 14:38, 29 October 2011 (UTC)

It's changed a couple of times since, the last by me. I've changed it back to natural number as not only is it more specific but it's less technical to call them 'numbers': the readership of an article like this will be very diverse so we should avoid technical language as much as possible. For the same reason 'divides' is better than 'divisor' (more commonly called 'factor' in some places), while both 'divisor' and 'since' were overused.--JohnBlackburne^words_deeds 15:11, 29 October 2011 (UTC)

Overloaded and strained introduction

You don´t need composite number to explain why 6 is not a prime and what´s more, if you use the conception, then you are as close to circular reasoning as you can be. You are there.

Furthermore, the stereotyped expression: "On the other hand", is here not only misplaced but also disturbing. Ex: 9 is an odd number, but "on the other hand" 10 is not.... . What other hand? Please, leave off the phrase.

More. Skip the literary turn with synonyms etc. "Divisor" is as good as "divider". If you use "divisor" in the definition, then it´s consistent to use it in explanations.

Finally. "Integer" is more natural than "natural number". Ask prime school pupils!

At last, thanks to Quondum. Answer to your question: For integers, prime element is identical with prime number.

Rebecca G (talk) 13:13, 30 October 2011 (UTC)

Technically you don't need composite to define primes, but it is helpful to give it as the name of non-primes, especially as one way of thinking of primes is that they are all things that have no factors other than 1 and themselves: they are all things that are not composite. As for 'on the other hand' and using 'divided' as well as 'divisor' that is just clear writing: if a reader is not familiar with the term divisor (many students are taught that they are factors instead) it may help them understand. 'on the other hand' clarifies that is something different, much better than 'not' (which was inappropriately stressed), and the barely grammatical ", since it except 1 and 6 also".--JohnBlackburne^words_deeds 13:46, 30 October 2011 (UTC)

My question was intended rhetorically. Look at prime element - it gives the specific example of prime elements in ℤ, which include negative values. My intention is to keep the definition in keeping with the more general definition, which is achieved very naturally by specifying natural numbers rather than integers. I'm also not convinced that integers are as familiar to children learning about numbers as you suppose. My perception is that the concept of negative numbers, and especially as numbers for multiplication, is not as solid as one would hope by the age some might be considering divisibility.

I fully agree that the phrase "on the other hand" must go. The rest of the sentence needs some consideration, else I'd remove that phrase myself now. Which ties in with the mention of compositeness; which should be in the lead, just like one would define even and odd numbers together, but not as an incidental property of the 6 mentioned in the example. Thus a sentence defining composite numbers should perhaps be the second sentence of the paragraph, and the example should be the third. Quondum (talk) 15:36, 30 October 2011 (UTC)

I can't really see why this edit was reverted. It does not alter the meaning of what is said, and the use of i.e. is clunky here.--♦IanMacM♦ ^{(talk to me)} 12:25, 31 October 2011 (UTC)

Just as the edit summary is saying, I reversed it because "we don't want to define primes as non-composites here, but just give another name to non-primes." It did change the meaning of the sentence, from mentioning a fact to drawing a conclusion from it. Unfortunately it was re-reverted by an editor who didn't pay enough attention to the stated reason, and focused on the perceived literary qualities of the text instead. WillNess (talk) 17:39, 31 October 2011 (UTC)

It is satisfactory to see that the expression "greater than 1" seems to be established and that the misplaced "On the other hand" at last has been removed.

Still it remains to throw some overload into the water. Nobody needs composite number to understand why 6 is not a prime and furthermore, the use here of this conception is circular reasoning.

With or without the "i.e." it´s messy.

Rebecca G (talk) 14:48, 31 October 2011 (UTC)

In my edit the composite number was just mentioned as another name for non-prime. I don't see anything wrong with mentioning that. WillNess (talk) 17:39, 31 October 2011 (UTC)

No one said it was wrong, just that it wasn't phrased as well as it might be. Dmcq (talk) 18:23, 31 October 2011 (UTC)

All is good, it's got its own sentence now! :) WillNess (talk) 19:25, 31 October 2011 (UTC)

For good. Please don´t begin all over again. Nobody needs composite number to understand why 6 is not a prime. The poor reader needs two lines. That´s all there is. There isn´t anymore.

At last. Let the definition stand alone, don´t complicate this and throw the rubbish to where it belongs.

Rebecca G (talk) 12:22, 1 November 2011 (UTC)

Later in the lead there is the sentence There is no known useful formula that yields all of the prime numbers and no composites. If this sentence is to remain in the lead, then "composite number" needs to be defined earlier (it's not just about the number 6; the notion of a composite number is mentioned several times througout the article). It's just a question of what is the most logical place to do this.

Please try to keep the discussion focussed on the content of the article. We don't need expressions of frustration or random links to strange places. Thanks, Jowa fan (talk) 12:42, 1 November 2011 (UTC)

I agree. In addition not only should there be four paragraphs but they should be balanced and make sense as paragraphs. So the first paragraph has the definitions, and the centrality of primes to arithmetic. The second paragraph is on primality and prime tests. The third is on the infinitude and distribution of them. And the fourth is on other theories and applications. Chopping the first paragraph in half and merging it with the second breaks this structure, and so makes the whole lead less coherent and readable. There's no policy to point to for this as it comes under "write good, clear English", something that's especially important for the lead section.--JohnBlackburne^words_deeds 13:49, 1 November 2011 (UTC)

For what it's worth, I totally agree. WillNess (talk) 13:54, 1 November 2011 (UTC)

Now, when you have signed your contribution, I will change my censored comment from: "What brilliant bore wrote this?", to: When sticklers for clauses and paragraphs sweep away simplicity and logic.

PeggyCummins (talk) 13:39, 2 November 2011 (UTC)

Not more?

PeggyCummins (talk) 17:31, 1 November 2011 (UTC)

Section titles: use of "the"

For some time (see e.g. this version from March) this page has had sections called "The fundamental theorem of arithmetic" and "The number of prime numbers". Recently someone deleted the word "the" from these headings (and from the newer section heading "The Zeta function and the Riemann hypothesis") citing Wikipedia:MOS#Article_titles. I restored the status quo, giving what I thought was a reasonably clear edit summary, but it has been reverted without any further explanation.

For the first and third instances, the relevant part of the MOS is Do not use A, An, or The as the first word...unless by convention it is an inseparable part of a name.... In this case I believe it is indeed part of the name: mathematicians don't say "by fundamental theorem of arithmetic" (without "the") any more than people say "I enjoyed reading Great Gatsby" (without "the"). In the case of "(the) number of prime numbers", I think this is a case where WP:IAR is justifiable: I simply think "number of prime numbers" looks silly. (And yes, we have a page called Fundamental theorem of arithmetic (without "the"), and a few other fundamental theorems. I think all of these pages ought to be renamed too, but that's a much bigger issue than changing some section headings, so I'm not about to get into that here.) Jowa fan (talk) 11:46, 19 December 2011 (UTC)

People don't say "I work at BBC", but the article is at BBC. And that's just the first article I tried. Or more relvantly it's Riemann Zeta function not The Riemann Zeta function. The naming conventions are clear and the reasoning is straightforward: why prepend 'The' or 'A' when it adds no meaning an makes the headings longer and so the TOC more crowded. They are headings not grammatically complete sentences or phrases.--JohnBlackburne^words_deeds 12:08, 19 December 2011 (UTC)

There is also a rule about capitalizing "the" when it is part of a name inside a sentence, and by this criterion it is not part of the name since it does not get capitalized. For that matter, the whole phrase is not treated as a proper name at all even though it is used like one; it is pretty normal to write "the fundamental theorem of arithmetic". The case for keeping the "the" seems rather marginal to me. On the reverts, I agree that more of an explanation would have been helpful; repeating a previous point already made is not very helpful. — Quondum^t_c 12:22, 19 December 2011 (UTC)

Thanks for the replies. Okay, if there really is a consensus regarding "Fundamental theorem..." and "Zeta function" then I won't argue further (although it's odd that the old version persisted for so long). But what about "Number of prime numbers"? Jowa fan (talk) 13:27, 19 December 2011 (UTC)

I think this is the whole point of that aspect of Wikipedia:MOS#Article_titles: to set a specific "style". It doesn't sound so silly when you get used to it in general. It does sound more stuffy and formal ("encyclopedic"), but quite frankly, there is far too much chatty pedagogy in Wikipedia for my liking and a more terse tone should be striven for if only to discourage this tendency. Wait a while – once you start expecting unadorned noun phrases as headings you might find it sounding more "normal" – even "Number of prime numbers". — Quondum^t_c 13:55, 19 December 2011 (UTC)