Talk:Prime number/Archive 7

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Tagging

The article has been tagged as "too technical" To be fair, it is no more technical than some other mathematics articles. To prevent drive by tagging, there should usually be an explanation of what the problem is, and how to improve it. What is wrong here?--♦IanMacM♦ ^{(talk to me)} 10:04, 24 February 2011 (UTC)

I also don't quite understand it. I asked the IP to explain his/her qualms here. Jakob.scholbach (talk) 12:00, 24 February 2011 (UTC)

I too saw this, and didn't think the article was over technical. If no explanation is forthcoming soon I think the tag should be removed. JamesBWatson (talk) 13:03, 24 February 2011 (UTC)

Why does this matter so much?

Other then being an interesting problem, why are spooky problems surrounding the prime numbers so darn important? —Preceding unsigned comment added by 207.159.173.37 (talk) 18:52, 7 April 2011 (UTC)

The least spooky (??) application is probably public key cryptography. That is, you could not do online banking without modular arithmetic, which in turn relies on primes. (To be fair, though, for this application, an awful lot of more sophisticated things, most of all elliptic curves, also need to be understood.) Jakob.scholbach (talk) 17:59, 9 April 2011 (UTC)

6 is not composite?

"6 is not a prime (or composite)"

Why not? — Preceding unsigned comment added by JohnJSal (talk • contribs) 20:56, 21 June 2011 (UTC)

I've changed it. Someone rewrote the lead a week or so ago and this crept in. Hut 8.5 20:59, 21 June 2011 (UTC)

Primes as building blocks of numbers

I find this statement confusing: "primes are numbers' building blocks". Everyone knows it is 0 and 1 that are the building blocks. Of course the intent is that it is w.r.t. multiplication that primes are mentioned; I propose clarification is in order: "primes are building blocks of numbers w.r.t. multiplication, just as 1 is, w.r.t. addition". What would be the preferred wording?

My recent edit in this matter was summarily reverted; I'd like to see some responses before I attempt to re-instate it (which I still think is appropriate). WillNess (talk) 09:08, 18 July 2011 (UTC)

I see no reason to reinstate it. The original form is quite clear. Everyone certainly does not know it is 0 and 1 that are the building blocks, and in the context there is no way of reading the statement as applying to anything other than multiplication.JamesBWatson (talk) 10:45, 18 July 2011 (UTC)

Aren't we (all) taught that any number is a sum of its predecessor and 1, first of all? Addition much precedes multiplication, is my impression. For me, the statement is confusing, as it is. I'd rather prefer not to guess the intent from the context, but be given the facts openly and straighforwardly. Especially that the needed addition is just a few words. WillNess (talk) 14:16, 18 July 2011 (UTC)

The reference could also be made in that respect to the fact that "the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the prime numbers"; and for natural numbers with addition the generators set is of just one element, the number 1. (from Free monoid) WillNess (talk) 10:10, 18 July 2011 (UTC)

What on earth for? Why turn a straightforward explanation of an elementary mathematical concept into something that is comprehensible to almost nobody other than graduate mathematicians, and not even all of them? JamesBWatson (talk) 10:45, 18 July 2011 (UTC)

As we go along the article, it gets more and more complicated. I specifically proposed to add a reference, that would appear far below; yet it would provide a further pointer for the interested to explore. Convincing enough? :)

Generating set is very simple and intuitive actually: we get all the naturals by adding a number of 1s. We also get them by multiplying a number of primes. It is in this sense that they are, the building blocks. WillNess (talk) 14:16, 18 July 2011 (UTC)

Primality of 1

As my last two edits have been reverted, I thought it would be better to ask before redoing yet another edit. Since the last revert seemed to suggest that stating prime and irreducibility is repeating another section later in the article. Would it be acceptable to simply state that 1 is a unit and therefore does not qualify under the algebraic definition of a prime? It seems the talk in the paragraph prior could be easily replaced with a single concise reference to the unambiguous algebraic definitions. Crasic (talk) 00:12, 30 June 2011 (UTC)

Just to Clarify, I would not be restating the definition of prime and irreducible (as in my previous edit). Something along the lines of " Since the number one is a unit it does not satisfy the definition of a prime. This justifies omitting it from the list of prime numbers " Crasic (talk) 00:14, 30 June 2011 (UTC)

I reverted you since, I think talking about prime elements and units this early in the article is neither necessary nor helpful for lay readers. We target an audience of 10 year old kids at this point in the article. Jakob.scholbach (talk) 13:59, 30 June 2011 (UTC)

I understand that, and in hindsight I agree that defining prime/irreducible elements was out of place in this early section. However a sentence mentioning that 1 is a unit would do a lot to justify the entire argument of the section (and refer to the relevant article/section for further information). Crasic (talk) 20:04, 30 June 2011 (UTC)

I disagree. Mentioning that 1 is a unit is not going to help readers who don't know rings, and in the interest of a lean presentation I prefer not putting any unnecessary interal cross-references. Also, the failure of the fundamental theorem of arithmetic (if 1 would be prime) is justifying the exclusion of 1 as a prime just as well (but more concretely) than the remark that 1 is a unit. The Lasker-Noether theorem also holds up to units, only, so the remark you want to put is just restating in more advanced language what is already clear at the elementary level. Jakob.scholbach (talk) 10:07, 1 July 2011 (UTC)

The article is about prime numbers. The overwhelming majority of people reading it won't have a clue what a ring is, and will be reading it to find out about prime numbers. Putting stuff in which is not on that topic will serve only to confuse them. Anyone who knows enough about rings to want to find out about such concepts as prime and irreducible in the ring context is much more likely to be looking for such information in articles about rings than in an article about prime numbers. Consequently including the information here is unlikely to be useful to anyone, and is likely to be a positive hindrance to some readers. JamesBWatson (talk) 11:33, 1 July 2011 (UTC)

Revert on 2011-09-24 by PrimeHunter

In reverting my recent addition,

"If 1 were to be admitted as prime there could exist no other prime numbers, with composites being non-trivial (i.e. with a coefficient greater than 1) multiples of a prime by definition, as any positive integer greater than 1 is a non-trivial multiple of 1. It can't be composite either, because a non-trivial multiple is by definition greater than 1."

PrimeHunter writes: "I think this creates unneeded confusion. Nobody suggests a definition where 1 is the only prime, and I haven't seen this argument in sources".

Well, I didn't suggest a definition where 1 is the only prime either. What I wrote was that if you were to admit 1 as prime, you couldn't have any other number as prime, since it would be a prime's (i.e. 1's) multiple (with coefficient greater than 1), i.e. a composite number. This has got to be the most direct, simple and easily understandable argument against the primality of 1 (unlike mentioning "several other properties" etc. as it is done in the article, which is unclear and confusing).

As written the subsection is long-winding and complex, while the argument I propose is simple, concise and direct.

As for sources, any source on sieve of Eratosthenes states that composite is a (non-trivial) multiple of some prime. For instance Atkin and Bernstein cited in the article states that "The idea of the sieve of Eratosthenes is to enumerate values of the reducible binary quadratic form xy". I.e., every composite is a (non-trivial) multiple of prime. We are allowed to interpret sources in a straightforward way are we not? WillNess (talk) 08:23, 25 September 2011 (UTC)

I agree with PrimeHunter - the text you provided is difficult and confusing to interpret, whereas the text preceding it is straightforward. Language issues aside, your statement is an attempt at a theorem/lemma proof that does not prove anything. Whether 1 is a prime number is dependent on exactly how primes and composites are defined. There is no contradiction either way, but all assertions and theorems using the concepts would have to take the exact definitions into account. It is a matter of choice, with the "cleanest" choice of definition having been chosen. Quondum (talk) 10:03, 25 September 2011 (UTC)

Language can always be improved upon and I would only welcome and in fact expect any suggestions from the community to that effect. What I wanted was just a little insight. The subsection text as it is, gives me no insight whatever. Yes 1 is defined as being non-prime, so why at all have a subsection trying to explain the reasons for it? Is it not for the insights why? WillNess (talk) 11:04, 25 September 2011 (UTC)

You are perfectly right on the point about language. The point of the section seems to do two things: give a history of how 1 was treated in the context, and to give a rationale behind the choice of the current definition that excludes 1 as prime. And you are saying that it does not achieve this second goal very well. It does this by means of an illustration of the added awkwardness when used for the uniqueness of factorization, and makes the point that this is not the only example. Any rationale for the choice to exclude 1 as a prime would probably have to follow this formula: if 1 was a prime, the theorems would look like such, with the following loss/gain of mathematical elegance or applicability - essentially arguing that excluding 1 produces a set that is more useful in the majority of contexts. If the current illustration is not enough, one might want to add another, though I find the existing rationale perfectly adequate, and am not sure what insight you are looking for that it does not provide. Quondum (talk) 11:44, 25 September 2011 (UTC)

Yes exactly, on the 2nd goal - it does so in a roundabout way, and my argument, IMO, gives a clear-cut straight answer: if 1 were prime, there could be no other primes, period. On the point of uniqueness of prime factorization, one could ask, so what? etc., so this would involve additional concepts, i.e. be more complex to comprehend. It's not as clear-cut and direct, for me, i.e. lacking in insight. And yes, this is exactly why I wanted to add this to the "current illustration". WillNess (talk) 12:42, 25 September 2011 (UTC)

I strongly disagree with the point you are apparently trying to make: "if 1 were prime, there could be no other primes, period." 1 has been defined as prime in the past by eminent mathematicians. All other primes were the same and they could still do advanced number theory. The definitions at the time were simply formulated to include 1 without excluding other current primes, for example "A prime is a positive integer which is only divisible by 1 and itself". If nobody has ever suggested a definition where 1 is the only prime then Wikipedia should not imply that the current definition is excluding 1 in order to avoid a fictional problem of 1 being the only prime. PrimeHunter (talk) 00:43, 26 September 2011 (UTC)

You've quoted my remark out of context. It was certainly not intended to be included in the article in that form. It was made as an attempt of succinct reformulation of the original text I proposed to include in the article. And that context was, "with composites being non-trivial (i.e. with a coefficient greater than 1) multiples of a prime by definition," ... "there could exist no other prime numbers" ... "if 1 were to be admitted as prime." Which is obviously a true statement. It is not claimed to be so in the context of any other complications to the basic definitions which might have been employed at times past by eminent mathematicians, but in the context of currently accepted definitions of primes and composites, from which the constructive definition of the sieve of Eratosthenes follows of primes as non-composites and composites as non-trivial multiples of primes, which is widely accepted as equivalent to the usual definition of primes as integers above 1 with singleton factorization and composites as non-primes. WillNess (talk) 15:11, 26 September 2011 (UTC)

(unindent) I concur with PrimeHunter's revert of your addition: it is quite confusingly written, and I doubt this explanation appears anywhere in the literature. If you can come up with a neat and concise formulation (e.g., what do you mean by coefficient?) and back it up by a reference, then we can discuss the matter again. Until then, I think it is better left as it is. Please keep in mind that is not our job to come up with explanations that we (editors) consider easy or convincing, but need to reflect the thoughts of scholars on the matter in question. The explanation you give strikes me as a little bit idiosyncratic and I have nowhere seen it before. Jakob.scholbach (talk) 15:32, 26 September 2011 (UTC)

Multiple (mathematics) defines coefficient in this context. As to the exclusion, no contest. :) As for "idiosyncratic", to what specifically are you referring (if you could tell me please, perhaps on my talk page)? As for its being "confusingly written" I'd expect help from the community, as we discussed above, if the idea itself were to find favor, as well as telling me if there's any need for sources or not (you already did). It's more than uniqueness of prime factorization: if primes are the integers with singleton prime factorization, any number above 1 would have non-singleton prime factorization, p = p · 1, so there would be no other primes, is what I'm saying. Cheers. WillNess (talk) 16:04, 26 September 2011 (UTC)

That's the basic problem. It's what you're saying. Not what any cited source says. And I agree wholly with everything else Jakob.scholbach says above. Dmcq (talk) 20:15, 26 September 2011 (UTC)

I wasn't disagreeing with what he said either. As for sources, it is my understanding that we don't need sources for trivial arguments and I thought it was a trivial argument. WillNess (talk) 08:07, 27 September 2011 (UTC)

Of course many of these things are trivial. It is, though, helpful and IMO necessary to stick to explanations that occur in "mainstream" textbooks etc., since this is what we can refer to. (The article currently does not have too many citations, but is written in a way such that most things would be easily citable.) Even if, say, there was only a very cumbersome definition of an elementary notion, we should not try to simplify it on wp, but need to present the state as it is. (For example, if every textbook said: "A prime is a number n that is either two or such that Z/nZ is a domain (mathematics) with an odd number of elements", we should stick to this definition). Jakob.scholbach (talk) 10:02, 27 September 2011 (UTC)

P.S. Maybe idiosyncratic was the wrong word. What I meant may be better described by "unusual argument". Somewhat in a similar vein, your point below about divisors goes in a similar direction: you have a certain idea that you consider simpler/more direct, but most other editors feel that consensus is to stick more closely to notions/explanations that we can lean on. So to speak, writing WP does not require a lot of creativity when coming up with mathematical arguments. Jakob.scholbach (talk) 10:02, 27 September 2011 (UTC)

Divisor is a mathematical concept, reflective of a physical process of dividing up a set of items into smaller sets of items (stones, apples, etc.). The physical process is primary, original; the mathematical concept is secondary, reflective. (You get a pile of items (a pile is more than one item); you try to subdivide it into equal-sized piles of items. If you can't, it was a prime. That is also why 1 is not a prime and is not not a prime. It is not a pile). There, you can teach it to a 5 years old.

And this I find extremely amusing: the exact same reference used to prop up the paragraph we discuss here (the other one being just someone's personal page), opens with a statement of 1 being "the unit (the building block) of the positive integers". For a context you need only look down a notch to the next thread on this page.

So this is all connected: 1 is included w.r.t. addition, and excluded w.r.t. multiplication. 1+x reflects a process of adding an item into a pile; 1*x doesn't reflect any process. With primes used as "building blocks" w.r.t. multiplication (as I tried to insert there into the article) including 1 in primes makes no sense: 1*x produces nothing new. But that's just an aside. This paragraph is speculative anyway. It's not like there was a commission on the exclusion of 1 from primes whose proceedings we could cite. Both references are essentially blogs. Demanding sources exclusively from me, and on such a basic and simple argument (p = p · 1) seems unreasonable in light of this. The only acceptable objection here is no consensus, and that's fine with me, even though I find it, well, idiosyncratic. Cheers. WillNess (talk) 10:08, 28 September 2011 (UTC)

As I said, the article is not in perfect shape and the references you refer to not so good either. (I once invested some time in writing the article, but did not get as far as the references.) However, given that all editors in this discussion disagree with your edit (but not with the current version of that paragraph), it seems reasonable to ask for a citation to build some sort of consensus. The closest guideline for the situation at hand is WP:CHALLENGED, I think (keeping in mind that nobody is challenging the truth of your argument, but it being used in any kind of mainstream textbook etc.) Jakob.scholbach (talk) 12:06, 28 September 2011 (UTC)

Definition and examples

I've added another description of a prime as "a number n > 1 that cannot be divided into a number of subparts of equal sizes". The wording is not very precise, but it should be OK to use imprecise wording when it makes the intent clearer. The more precise statement would be "... can't be divided into any number of equal subparts" but that reads as somehow less immediately clear to me. Would appreciate any improvement.

My intent is to state it in the simplest terms, e.g. when children first learn about numbers through counting sizes of piles of stones (or apples or whatever). A pile's size, expressed as a number of stones in it, is prime when a pile can not be divided into any number of equal-sized sub-piles of stones. I think this makes an intuitive and clear a picture of what "prime" means. Speaking of "divisors" is much less intuitive.

The picture in the article in that same section also tries to subdivide a number into rows of equal sizes. Some wording should appear in the article along those lines IMO. The goal is to make it as simple and clear as possible, for as big an audience as possible. WillNess (talk) 09:38, 18 July 2011 (UTC)

I appreciate your intention, but I don't think the sentence you added is helpful. "a number n > 1 that cannot be divided into a number of subparts of equal sizes" is suboptimal in a number of ways: what does divide mean (n=a+b, n=a*b)?, "a number" is vague (>1?), "subparts" is non-standard, "equal sizes" is just wrong, the divisors don't need to have equal size. I'm going to revert this addition. Please don't take it as an offense in any sense, the sentence is just not an improvement of the article and I don't see a simple rewording that would remedy all its problems. Jakob.scholbach (talk) 10:45, 18 July 2011 (UTC)

OK, the wording wasn't clear, that's why I asked for help with it, and for opinions. I'm not talking about 'divisors' here; i'm talking about physically dividing a pile of apples into several equal piles of apples (n = a+a+...+a). That would directly correspond to the n = a*b above it, in the article.

I want to make it as intuitive as possible. Talking of multiplication operation, and division operation, and remainders, is less intuitive, and could be avoided with this idea.

So what could be the wording? The correct formulation of the same idea, that with numbers seen as cardinalities of finite sets (which is what they are after all, the amount of stuff in a pile as we count it, right?) a prime number corresponds to a set which can not be subdivided into any number of subsets of equal sizes, was rejected as "too complex". For me the insight of dividing a pile into equal subpiles makes the whole picture much clearer, and accessible to an 8 year old. Again I draw your attention to the picture above the section in the article, which is left unexplained in the text. What can be done here? WillNess (talk) 14:48, 18 July 2011 (UTC)

What useful information does the change add to the account already given? I can't imagine anyone who doesn't understand the existing definition but who can understand this roundabout description. Is there really a likelihood that the article will be read by people who don't understand what "divisor" means and cannot work out what it means from the examples given, but can grasp the meaning of "divided into any number of equal subparts"? JamesBWatson (talk) 10:53, 18 July 2011 (UTC)

You may find this incredible, but many non-native English speakers read English Wikipedia, and for them "divisor", "dividend" etc. are all foreign. Same with 8 year old children. Speaking of dividing sets (which is after all a plain English word first, and mathematical concept second) of stuff into equal-sized sub-sets is simple enough for an 8 year old, and clears up a picture significantly. Or else what would be the purpose of the picture in the article showing attempts at dividing a bunch of little squares into a number (any number) of equal-sized rows? WillNess (talk) 14:48, 18 July 2011 (UTC)

It does sound much better than the original bit about sets and cardinalities but it does have problems like that the n>1 has to be associated with the subparts instead. The major problem it seems to be trying to deal with is the word 'divisors' in the lead. Should we amplify on the term rather than just referring to the other article? Perhaps it is enough to say something in the first illustration like that a prime number can't be arranged as a rectangle with both sides > 1? Dmcq (talk) 14:28, 18 July 2011 (UTC)

Thank you, I'm trying. :) How about, "a number n > 1 is prime if for any set with n elements in it, it is impossible to place those elements into any number (greater than 1) of equal-sized new sets"? Or "it is impossible to partition that set into (more than one) equal-sized sub-sets"?

BTW have you looked at the other section above this one? Does it make you change your mind about it? WillNess (talk) 15:02, 18 July 2011 (UTC)

Well, you have to exclude the n-lots-of-1 case as well as the 1-lot-of-n case. Gandalf61 (talk) 15:32, 18 July 2011 (UTC)

Saying "n > 1" and "more than one", I thought I did. WillNess (talk) 16:50, 18 July 2011 (UTC)

... And then the whole sentence becomes very clunky. That's what I was avoiding by saying both sides > 1. We don't have sources talking like that so it could only go in as an informal explanation that helped significantly to explain the concept. I'll try out an extended description on the illustration and see if people fire it down. Dmcq (talk) 15:48, 18 July 2011 (UTC)

I think that last variant was OK: "a number n > 1 is prime if any set with n elements in it can not be divided up into equal-sized sub-sets". That is an English translation of the impossibility of n = a*b which appears in the text. Sources, what sources. (just kiddin'). I would much rather see some explanation in the body of the text than hidden away in small script under a picture. WillNess (talk) 16:50, 18 July 2011 (UTC)

As has been pointed out a few times if the sub-sets can be of size 1 then no number is a prime. Dmcq (talk) 16:58, 18 July 2011 (UTC)

Willness, our target audience is people that don't know maths, but not ones that don't know English. A 8-year old kid would hardly look up a mathematical concept in a foreign language. If so, they might first try the simple English version. The treatment could be made more accessible maybe by first coming up with two concrete examples, 5 and 6. Then, explain divisors. Finally, give the definition of prime as it is now. Then, reformulate that definition (impossible to write it as a proper product, exactly two divisors). Jakob.scholbach (talk) 18:01, 18 July 2011 (UTC)

We don't need to know what "divisors" are, that's the whole point about it. How about, "a number n > 1 is prime if n items can not be divided up into smaller equal-sized groups of more than one item"? WillNess (talk) 08:46, 19 July 2011 (UTC)

Well one can certainly get round mentioning the word divisor like in the first illustration in the article, but there isn't a great deal of point to that in the text. Personally I'm not a fan of school teaching which turn maths into biology with loads of names for different types of triangles and how to tell if they're the same, but even with that bent 'divisor' is a very useful word and concept and is always used in relation to primes. Wikipedia is supposed to just go with the herd and describe what is out there, that's a major part of its brief. Dmcq (talk) 15:07, 27 September 2011 (UTC)

Lead sentence and negative divisors

Until recently (June this year), the lead sentence of the article was

A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself.

Then it was changed to the more elegantly phrased

A natural number is called a prime number (or a prime) if it is greater than one and has no divisors other than 1 and itself.

Today someone inserted the word positive before divisors, and this edit was reverted. I think there's a problem here. The sentence does start by establishing the context of natural numbers, and so we can hope that people will deduce that "divisors" refers only to natural numbers. However, if someone follows the link to divisor, that page makes it clear that negative integers can also be divisors. So, according to what is written, there are no primes (because all natural numbers have -1 as a divisor). I think one way or another we do need to make it clear that divisors in this context are assumed to be positive. Jowa fan (talk) 07:39, 8 October 2011 (UTC)

I have restored the pre-June definition - it was both simpler and clearer than the new version. It also avoids any ambiguity over whether "if" means "if and only if" - see WP:MOSMATH#Writing style in mathematics. Gandalf61 (talk) 09:11, 8 October 2011 (UTC)

Yep the > 1 was necessary - sorry

Quite right David Eppstein, the > 1 is necessary. Very silly of me, and I'd complained above about people not being careful about the definition. Just because it would look nicer without extra conditions doesn't make it so. Dmcq (talk) 11:53, 17 October 2011 (UTC)

What are the 9 Diophantine Equations?

In 6.1, the last paragraph mentions a system of Diophantine equations. How can I find them or what are they? (A citation after that comment would be nice. I'll add one, if I can find a resource that has them.) — Preceding unsigned comment added by 50.55.212.199 (talk) 01:58, 21 October 2011 (UTC)

See formula for primes. Note that this article does not say there is a system of 9 equations; rather, it is a system of equations in 9 variables. —Mark Dominus (talk) 02:51, 21 October 2011 (UTC)

A correct, and attractive, definition of a prime number.

The current definition is ugly. It is not for mathematicians and not for non-mathematicians. ":1 and itself" is not pleasant and it´s also redundant. As a whole, the definition is a mishap.

"My definition" is not an innovation of mine. It is classic.

Furthermore, let the definition stand on a separate line.

Don´t tell me that there are as many definitions as there are people. But there are many "definitions" of the Quondum-type.

PeggyCummins (talk) 12:44, 23 October 2011 (UTC)

The current definition is straightforward, well written and about as clear as it can be. The one you added,

"A prime number p, is a natural number greater than 1, whose only distinct natural number divisors are 1 and p."

unnecessarily uses symbols to express something easily expressed in natural language. Having a paragraph that consists of only a single short sentence is bad writing. And lastly that's three editors that have restored the version before your change, so there is a clear consensus to keep that. Before you make any other change please establish a consensus for it on this talk page.--JohnBlackburne^words_deeds 13:08, 23 October 2011 (UTC)

I agree with JohnBlackburne that you should refrain from edits until consensus has been reached, and about the unecessary extra symbol. However, single-sentence paragraphs provide emphasis; this is "bad writing" when such emphasis is excessive or inappropriate, and should be used very sparingly. Quondum (talk) 13:44, 23 October 2011 (UTC)

Now I understand where from the "natural and straightforward" Wiki-formulation of the Pythagorean Theorem originates.

I am sorry, but the current definition of a prime number is badly written and not straightforward. It is only "about as clear as it can be", but not clear.

Do you think that the sympol "p" in the classic, correct and clear definition is very hard and troublesome?

Having a paragraph that consists of a single definition, is clear and good writing. Where did you two squeakers study?

At last; The "definition" given by Quondum, that is a fine example of "bad writing".

PeggyCummins (talk) 16:07, 23 October 2011 (UTC)

I think that this article is going to be read by primary school students who are not yet comfortable with algebra. So, yes, I think having the algebraic symbol p in the very first sentence is a problem. —David Eppstein (talk) 17:32, 23 October 2011 (UTC)

I agree. While parts of this article are inevitably going to be inaccessible to readers who do not have some mathematical knowledge the lead should be accessible to everyone. Pythagoras' theorem can't really be stated without using either symbols or words that the average reader may not understand, but that's not the case here. Hut 8.5 19:39, 23 October 2011 (UTC)

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)

Gross omissions

The article does not include the contributions of Indian mathematicians, particularly the ancient ones, to prime number theory. Prime numbers and many of their properties were well known thousands of years ago in India, even before the Greeks got to them. A simple google search will reveal just that. The article as it currently stands is incomplete and misleading.

Thanks 173.56.80.54 (talk) 13:25, 28 April 2012 (UTC)

I didn't find anything specific in a brief Google search. If you want something added then you should provide sources and not ask others to find them. PrimeHunter (talk) 13:36, 28 April 2012 (UTC)

misleading edit

This was in some respects quite an unfortunate edit and I regret that I didn't fix it until today. Calling the members of "any finite set of primes" p₁,...,p_n is misleading because that notation is conventionally used to denote the first n primes, rather than an arbitrary finite set of primes such as {5,7}. Then it says that their product plus 1 is called a Euclid number, which would make 36 a Euclid number since 5·7 + 1 = 36. But the term "Euclid number" in conventional use means something you get by multiplying the smallest n primes and adding 1. Then later it says this number need not be prime. Indeed, 5·7 + 1 is 36, and that is far from prime. But the kind of example added here makes one think this is about Euclid numbers as conventionally defined. Michael Hardy (talk) 18:28, 30 May 2012 (UTC)

I think you actually mean this edit. But in any case the point is the same, and the new version is better. Jowa fan (talk) 00:25, 31 May 2012 (UTC)

Dual predictive Prime number chains to simplify the mystery of prime numbers(recent discovery)

WP:NOTAFORUM, articles cannot include original research.
The following discussion has been closed. Please do not modify it.
Dual Prime Number Chains:Cameron V, Denotter T. Prime Numbers 2012J Am Sci 2012:8(7):329-334).(issn1545-1003) Prime numbers are no longer considered a disparate mystery and much that may be written about Prime numbers is incidental to a rational prime number distribution by two chains that are modulated by a series of half-line numbers that seem to hold a spiral divergence and convergence as is shown in the cited manuscript. These examples are a no- brainer and are obviously and apparently infinite, although it has been demonstrated to hold true till prime 200 in the cited reference. Half- line numbers are in red, all divisible by 2 and their value advances in sets as 10, 20, 30, 40, 50 (10 at 1, 20 at prime 59, and 30 at prime 199) A. (P5*11)+(11*12)=(P11*17) > (P11*17)+(P17*12)=(P17*P23) >(P17*P23)+(23*14)=(P23*P31)…so on B. (P7*P13)+ (P13*12) = (P13*P19)> (P13*P19) + (P19*16) = (P19*P29) > (P19*P29) + (P29*18) = (P29*P37)…so on Reference: Cameron V, Denotter T. Prime numbers 2012. J Am Sci 2012; 8(7):329-334. (ISSN 1545-1003) 50.50.46.98 (talk) 08:32, 4 July 2012 (UTC) This is original research.--♦IanMacM♦ (talk to me) 08:41, 4 July 2012 (UTC) Journal of American Science. I guess to fool people looking for the American Journal of Science. That is called typosquatting when done on the internet. Dmcq (talk) 10:45, 4 July 2012 (UTC) The introduction of the paper [1] starts: "It is a most horrible fate not to be understood in science in spite of one’s best efforts. We have now written a very clear and simple manuscript on prime numbers to make for a simpler and easier fare than that of axiom 1:3 mathematics. We have stated and shown that current mathematics is in error, and that the current mathematical π is not the correct Pi value as referenced at the end of this manuscript. We will never take that statement back and never take back the new mathematics of Axiom 1:3 that we have discovered recently" So the author admits that their work is fringe theories. It doesn't belong in Wikipedia. If you are the authour then see also Wikipedia:Conflict of interest. PrimeHunter (talk) 11:09, 4 July 2012 (UTC) Yes, this is the same drivel that was being peddled by Vinoo Cameron (talk · contribs) and Theo denotter (talk · contribs) a few years ago. If I remember rightly, they claimed to solved the P=NP problem, squared the circle (although they didn't understand what the original problem was), and proved the Riemann hypothesis (though if their work was correct they would actually have disproved it). Hut 8.5 13:01, 4 July 2012 (UTC) This research paper also gives the correct value of Pi as 3.14159292035. Don't hold your breath waiting for mathematical journals to verify this result.--♦IanMacM♦ (talk to me) 13:14, 4 July 2012 (UTC) Archimedes' method with a 6225-gon suffices to disprove this. So I guess that gives an upper bound the the author's competence. CRGreathouse (t | c) 15:12, 5 July 2012 (UTC)

Don Blazys (prime generating) constant

What's about Don Blazys constant: 2.5665438321713888444675291063322857517829728287023146459697335254663997198904... (and the relationship between prime numbers and polygonal numbers of order greater than 2) which is able to generate all prime numbers [2]? 46.115.59.142 (talk) 13:45, 20 October 2012 (UTC)

Reliable source? --JBL (talk) 14:27, 20 October 2012 (UTC)

That's just the value of the continued fraction:

$2+{\frac {2}{3+{\frac {3}{5+{\frac {5}{7+...}}}}}}$

It doesn't "generate" prime numbers - you have to put them in to start with. You can make many such arbitrary "constants" out of the prime numbers or any other sequence of integers. Without a reliable third-party source, this just isn't notable. Gandalf61 (talk) 16:40, 20 October 2012 (UTC)

It is not impossible for such things to be notable -- see for example Mills' constant. The question is, what, if anything, do independent reliable sources say about this number? Deltahedron (talk) 18:15, 20 October 2012 (UTC)

Nothing. It gets 7 Google hits and no hits at all on Google Scholar or Google Books. Certainly doesn't warrant a mention here. Hut 8.5 20:18, 20 October 2012 (UTC)

Fair enough. Deltahedron (talk) 10:36, 21 October 2012 (UTC)

"...multiplicative identity."

Pardon my lack of expertise here, but I searched the topics and could not find a reference to the phrase "multiplicative identity." If you could break that down, or start another article on the subject that it may be referred to, this may aid in making this a Good Article. Wikipedia is the place where the layman can gather a comprehensive, albeit perfunctory (with respect to the knowledge base behind developing the article), understanding of a subject. The clearer the language, the more engaged the reader becomes, which benefits the article as well. MVD (talk) 10:41, 8 February 2013 (UTC)

I've added a link to Identity element. The phrase "1 is the multiplicative identity" is really a fancy way of saying that when you multiply a number by 1 it doesn't change. Hut 8.5 10:57, 8 February 2013 (UTC)

In the study of mathematical structures there are two important types which are usful in the study of numbers: Those are the additive and the multiplicative groups. A group is defined as a set of elements with a binary associative operation which is closed and defined over the set of elements. Also there should be a identity element which does not modify the other elements when operated. They are called additive and multiplicative identities, 0 and 1 respectively. A group should also have some elements, called inverses, which when operated with the corresponding element yield the identity element. Namely: If the whole positive numbers are the set of elements, denoted n, n = 1,2,3..., and + denotes addition, then the operation AOP(3,6)= (3 + 6) = 9; it is closed because 9 is also a whole integer. The additive identity is 0, OP(0,n) = 0 + n = n. and AOP(n,m) = ( n + m ) = 0 → m = -n → AOP(n,-n) = 0. To have a group then there is need to increment the set of whole integers by adding the identiy 0 and the negative whole integers. The additive group,AG(n,+),n is positive or negative or zero, + addition operation. + is associative because AOP(AOP(n,m),z) = [(n+m)+ z], since AOP(a,b) = AOP(b,a) it is also commutative, also called Abelian For the whole numbers the multiplicative identity is 1, and integer MOP(1,n) = 1xn = n. There is no inverse defined, unless the rational numbers are defined MOP(n,y) = nxy = 1 → y = 1/n; n≠0. MG(r,x ) where r is a rational and 0 is excluded, and x is the ordinary number multiplication.190.128.41.216 (talk) 00:18, 9 July 2013 (UTC)

Misattribution to Chinese

"A special case of Fermat's theorem may have been known much earlier by the Chinese."

A valid reference would be nice. I doubt one exists,so have deleted the line.Septimus.stevens (talk) 20:35, 24 March 2013 (UTC)

It refers to the Chinese hypothesis which apparently isn't as old as often claimed. PrimeHunter (talk) 00:23, 25 March 2013 (UTC)

Bounded Gaps

http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html Subzbharti (talk) 13:06, 24 May 2013 (UTC)

Numbers selected at random

Without a specified probability distribution on the integers, the statement about the likelihood of two numbers being coprime is meaningless. There is of course no uniform distribution, there are a variety of reasonable distributions, and an unreasonable distribution can be chosen to make the probability 0. eigenlambda (talk) 04:08, 6 September 2013 (UTC)

It's the limit as explained in the linked article coprime. It's often just called the probability with no further explanation and I don't think we need the technical details in this article but I have added an online reference which also explains it.[3] PrimeHunter (talk) 11:51, 6 September 2013 (UTC)

We need to add a hint that this particular information would be available via that link; being a deterministic concept, it is not a link one might follow if this question occurs to one. — Quondum 12:01, 6 September 2013 (UTC)

I have now added a link to the section with the detail. — Quondum 12:06, 6 September 2013 (UTC)

Definition

The definition currently offered is "A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself." which is not neat and often prone to confusion especially regarding the status of 1. A while ago, I also saw somebody arguing that 0 was prime :D Can we not instead define prime number as "one which has exactly two unique divisors"? isoham (talk) 16:41, 18 September 2013 (UTC)

Many years ago I used to prefer to define prime numbers as having exactly two factors, as it avoided having to make a special case for 1. However, I now think it much better that we use the more standard definition, for several reasons. Firstly, the very fact that it is the standard definition is a significant reason, as we should reflect what is normal in reliable sources, not make our own decisions to do something that we regard as superior to standard mathematical practice, which is a form of original research, albeit a rather trivial form. Secondly, 1 actually is a special case, quite different from composite numbers, and is not counted as a prime for completely different reasons from the reason why, for example, 12 is not regarded as prime. Better to use a definition which makes it clear that 1 is different, rather than a definition which artificially obscures that difference. Thirdly, the essential characteristic of prime numbers that gives them a special role in number theory is that they have no proper factors, while the fact that the number of factors they have happens to be two is not particularly significant. It is much better to give a definition that emphasises the feature which is significant and important, rather than an incidental characteristic of no special importance. Fourthly, the concept of a prime can be generalised to rings other than the ring of integers, and in that case there is no special role played by having exactly two factors. Of course most readers of this article will never venture into that area, but it is more helpful to give in the most elementary case a definition related to the generalisation, rather than an idiosyncratic definition which obscures the relationship between the two. As for somebody arguing that 0 is prime, that is completely irrelevant, since it is inconsistent with both definitions. JamesBWatson (talk) 13:26, 19 September 2013 (UTC)

Taking a Devil's advocate position, an irreducible in a ring is one that has exactly two (equivalence classes, with respect to multiplication by units, of) divisors. The fact that irreducibles are the same as primes over the integers is an interesting theorem. I quite agree that 1 (and 0) are special cases, but "exactly two divisors" does have a place in ring theory. — Arthur Rubin (talk) 16:32, 19 September 2013 (UTC)

OK, I accept that, if you replace "divisors" by "equivalence classes of divisors", then something akin to "exactly two divisors" does apply in rings. I also glossed over the fact that the usual definition of primes in the natural numbers is by convention taken as the definition of "irreducible element" in other rings, rather than of "prime". For both those reasons, my appeal to more general ring theory is perhaps not as good a justification of sticking to the standard definition as I thought when I wrote it, but I still stand by the other reasons I gave. Of course "exactly two divisors" does not apply even in the ring of integers (as opposed to the natural numbers) unless we take "divisor" as referring to equivalence classes. I have also thought of another reason for disagreeing with the proposal to replace the definition. The justification given is that the standard definition is "often prone to confusion especially regarding the status of 1". Indeed, that was one of the reasons that I used to use the "two divisors" definition years ago. However, from experience, I found that people who were taught that definition were just as likely to think that 1 was a prime as people who were taught the standard definition. I have no idea why that is, and it seems totally counter-intuitive, but it is an empirically observed fact, and so the apparent advantage of the alternative definition simply does not apply. JamesBWatson (talk) 20:06, 19 September 2013 (UTC)

Ending figure ?

Obviously, there are only one prime that has the ending figure of "2" or "5" - 2 and 5 themselves. And no prime ends with 0,4,6 or 8. So with exception of the numbers 2 and 5 all other primes end has an ending figure of 1, 3, 7 or 9. The total amount of all primes are indefinite. But are there an (approx.) equal amounts of primes ending with 1, 3, 7 or 9 , whithin a certain range ? Boeing720 (talk) 00:19, 17 October 2013 (UTC)

Yes; see the "remarks" section of Siegel–Walfisz theorem, which gives a formula for the number of primes with different remainders (mod 5 or mod anything else) where the leading term of the formula doesn't depend on the remainder. —David Eppstein (talk) 00:44, 17 October 2013 (UTC)

Appriciated and thanks for Your effort to make me understand. However I'm sorry to say that I don't fully understand it all, for instance this sign $\equiv$. Is it not equal or what ? (In Sweden when I red math, the sign for not eaqual was an eaqual sign with a slash across. The versal sigma-sum sign I do understand though. I'm also very familiar with mod and remainder. 17 mod 7 = 3 since an integer division 17/7 = 2 with a remainder of 3. I'm making this example only due to my own uncertainty) But, if possible, do You know if it's proven that the ending figures of primes are (fairly) evenly distributed between 1,3,7 and 9 ? The opposite would then be, if primes with an ending figure of for instance 3, are more common than the ending figures 1,7 or 9 (as an example). Boeing720 (talk) 18:31, 17 October 2013 (UTC)

Visualization

Imagine choosing a number on the number line insuring that the wave frequency of every past number or combination thereof does not intersect the axis where you choose. (the progression depicted doesn't show combination waves per say)

File:Sawtooth-intersections.jpg

sawtooth intersections, chinese remainders

— Preceding unsigned comment added by 72.219.207.160 (talk) 20:56, 14 December 2013 (UTC)

Reader feedback: why isn't 1 a prime?

70.169.106.37 posted this comment on 2013-12-07 (view all feedback).

When I was a boy in the 1960s, 1 was included as a prime number. I'm curious when, and how, the "greater than 1" part of the definition came to be?

Good question! A section on "Primality of one" was recently added. ★NealMcB★ (talk) 06:35, 3 January 2014 (UTC)

Recently? It may have been added in the current decade, but I doubt it. — Arthur Rubin (talk) 21:09, 3 January 2014 (UTC)

Semi-protected edit request on 14 May 2014

Add the prime number 57 to the '1st 100 prime numbers' section, it is missing.

99.242.224.75 (talk) 15:07, 14 May 2014 (UTC)

Er, what is 19 X 3?--♦IanMacM♦ ^{(talk to me)} 15:34, 14 May 2014 (UTC)

About the P

I love you, P!

semi-protected edit request

Please extend list of primes less than 1000:

The list of primes less than 1090: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087 (181 primes)

— Preceding unsigned comment added by 83.6.59.234 (talk) 07:08, 20 June 2014 (UTC)

Why? —David Eppstein (talk) 07:29, 20 June 2014 (UTC)

Because i use hexadecimal system, not decimal. 2^10=1024, 10^3=1000. Some people may be interesed on primes beyond 1024. — Preceding unsigned comment added by 95.49.26.86 (talk) 12:00, 5 July 2014 (UTC)