Talk:Prime number/Archive 5

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Exploration of primes in other bases

I notice another question above about using other bases. I understand that of course primes are not base-dependent. However I would like to learn more about exploration of primes in other bases. Can someone recommend a place to visit to learn more (for a non-mathemetician who isn't afraid of math)? A couple things I'm wondering... First, does using base 15 (or perhaps 16? or 7.5?) shed any light on prime quadruplets? Second, does exploring non-integer bases shed any light on searching for patterns in primes? I'm sure mathematicians have explored this thoroughly, so I figure I might as well see what they found out rather than bashing my relatively-uneducated head against base conversions. Thanks! — Epastore 22:03, 22 August 2006 (UTC)

The integer bases where patterns would be most obvious would be the primorials (e.g. 6, 30, 210, etc.). You would notice, for instance, that in base 6, all primes except 2 and 3 end in 1 or 5, and in base 30 (using the letters A-T to represent the digits 10-29), prime quadruplets always end in {B, D, H, J} and the rest of the number to the right of first digit mod 7 is always 0, 3, or 6. But that's pretty basic stuff for anyone who's played around with prime numbers much. Hopefully someone else can give a more sophisticated response to your question. --Mwalimu59 22:22, 22 August 2006 (UTC)

Thanks for the input. I just wanted to point out that on further reflection, it occurs to me that since primes are a function of integers, using a non-integer base would change things pretty radically. Or would it? — Epastore 01:13, 23 August 2006 (UTC

I believe primes are by definition integers. The only non-integer Epastore mentions as a possible base is 7.5. I doubt whether there is any way to use a non-integer base to represent the integers. Exploring that theme looks like original research.84.210.139.189 14:20, 25 August 2007 (UTC)

Reasons for not promoting to good article

Hi all,

I have not promoted this article to the status of good article because I feel there are structural issues with the article. The article reads like a collection of information on prime numbers, not an article on prime numbers. My specific suggestions are to cut-down and focus the lead (for example, the ring theory and knot theory information could probably be placed elsewhere) and look at how to amalgamate some sections and remove other sections (Primes in pop culture could probably go altogether). Generally, try to get a good flow through the article and don't be afraid to leave some information out. Once you feel these issues have been addressed feel free to renominate the article.

Cedars 05:35, 19 September 2006 (UTC)

Thanks for the suggestions to improve the article. -Hyad 01:48, 21 September 2006 (UTC)

So-called Second Generation primes

I haven't seen this classification either, and am (was...) somewhat more familiar with classification- if one can call it that- by remainder modulo various powers of 2, generally speaking? Schissel | Sound the Note! 14:46, 5 October 2006 (UTC)

Talk page archiveing

This talk page is now over 100KB in size. Could someone who knows how please archive any inactive discussion. Tompw 14:30, 7 October 2006 (UTC)

Done. Jɪmp 07:30, 29 July 2007 (UTC)

Random series

It is often told that the prime numbers are randomly scattered over the naturals. Peahaps, anybody can elaborate what does it mean? Should there be a rule inducting whether the following n+1 number is prime or not? But we have such a rule. Does it mean that the test isPrime(N) must be somewhat (sub)linar O(N)? Is, for instance, the factorial series 1, 1*2, 1*2*3, ... a random one and what is the difference to the primes? Googling fot the meaning of "random", I have got this nice explatantion:

"It is evident that the prime numbers are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990, [1])

--Javalenok 14:24, 15 November 2006 (UTC)

I think "random" here means more of a perception than a mathematical fact. Choose a random group of 50 consecutive integers. Half of those will be even and half will be odd. At least half will be composite. How many will be prime? Anton Mravcek 17:05, 15 November 2006 (UTC)

I haven't seen anything sensible about "randomness" of primes which is claimed proven by a reliable source and suitable for the article. And I haven't seen an acceptable (for me) formal definition of "complete randomness" which the primes might satisfy. However, there are many published plausibly conjectured properties one might expect from primes if they are "random", but all these put together are not enough (in my opinion) to call the set itself 'random' - I basically agree with R. C. Vaughan. The prime number theorem (PNT) roughly says a randomly chosen number around n has around 1/log(n) chance of being prime, but many definitely 'non-random' sets also satisfy that. There are many unproven conjectures based roughly on the thinking: "If we assume primes thin out as PNT but otherwise behave 'randomly' (e.g. has 1/log(n) chance of being prime independently of eachother, except for known patterns of prime factors), then there are probabilistic arguments that we would expect them to have certain statistical properties in the long run". These properties can e.g. be about the expected number of certain prime patterns, as in the broad Hypothesis H with quantification. I often search prime patterns and they usually behave approximately as I expected based on probabilistic arguments, but that proves nothing in general. PrimeHunter 20:00, 15 November 2006 (UTC)

The theory of algorithmic complexity defines randomness as infinite complexity, which corresponds to infinitely long program to describe it. But the algorithm discovering all the primes is quite finite. Indeed, the sequence of primes is not random, since using this algorithm we can always induce next prime.

//The primes generating algorithm
next_p: for p = 2 to infinity {
	for q = 2 to sqrt(p)
		if p mod q = 0 then
			continue next_p;
	output(p);
}

The article Prime numbers not so random? in nature.com alludes the randomness to the lack of patterns. Indeed, more regularity means more patterns (or less patterns). Brrr. The less rules there are in the language including exceptions the simpler the language is the easier it is to figure out looking at its sentencies. Here, the grammar consisting of only one rule "there are no rules" is equivalent to infinite-rule (complex) grammar (like telling n natural is equivalent to n = 1,2,3,...). OK, peahaps by the randomness of a sentence they mean (im)possibility to figure out the grammar, the sentence-generating rule (looking at 2, 3, 5, 7, 11, ...)? --Javalenok 10:02, 21 November 2006 (UTC)

 n      | 1 2 3 4 5 6 7 8 9 10 11 12 13
 isP(n) | 1 1 1 0 1 0 1 0 0 0  1  0  1

G. Chaitin defines randomness via the algorithm complexity (size) generating the series [2]. I have shown in the previous post a simple algorithm generating the infinite sequence isP(n). Since such algorithm exists, the series is not random. What is wrong? --Javalenok 07:37, 7 February 2007 (UTC)

There are different definitions of "random", and the term is often used without a mathematical definition. Professional mathematicians have said things like primes "appear"/"seem" randomly distributed, so we shouldn't just dismiss it based on one definition of randomness. Assuming some level of randomness when working with primes often gives good results, e.g. when guessing the number of twin primes in an interval. The only "random" claim the article currently contains is:

"when looking at individual numbers, the primes seem to be randomly distributed, but the "global" distribution of primes follows well-defined laws"

I think that is acceptable: It says "seem" (implies a human perspective to me), it doesn't mention a randomness definition which the primes might fail, and sayings like this are common in reliable sources. The primes are exactly defined using finite space so they by definition fail some randomness definitions. Any well-defined notion of "randomness" would probably have to be formulated with primes in mind, in order to give the primes a chance of satisfying it. If the infinite sequence of primes has sufficiently many properties which have "heuristic probability 1" based on certain randomness assumptions, then calling them "randomly distributed" may be sensible. PrimeHunter 15:48, 7 February 2007 (UTC)

Twin prime conjecture

Is it not proved yet? I think this is already known. (???) Franp9am 00:50, 10 December 2006 (UTC)

It is not proved yet. Many amateur mathematicians claim to have proved it but their "proofs" are usually trivially false or nonsense. A few more serious mathematicians have also attempted to publish proofs with errors. No alleged proof satisfies Wikipedia:No original research and Wikipedia:Verifiability. See also Talk:Twin prime conjecture. PrimeHunter 03:04, 10 December 2006 (UTC)

Wording of section on Open Questions

My eyes landed on the following line in the article:

There are infinitely many Mersenne primes but not Fermat primes.

and was startled to see unresolved conjectures stated as fact . . . until I realized that I was reading from the Open Questions section, and that all declarative statements in this section are solely conjectures.

This misunderstanding leads me to think that it would be better to phrase all -- not just some -- of the Open Questions as questions, rather than declarative statements, to avoid future misunderstandings of this kind.Daqu 18:38, 20 December 2006 (UTC)

The Core

Should we mention that in The Core, one of the scientists is sent a coded message using prime numbers? This could go under the pop culture section. Bio 19:36, 20 December 2006 (UTC)

Executable prime numbers?

The current version of the Trivia section contains an astonishing statement: "Such numbers, when converted to binary and executed as a computer program, perform acts encumbered by applicable law in one or more jurisdictions." If they can indeed be executed as programs, this is quite strange and definitely needs a citation. More likely the text should say that the numbers will let you unlock the encryption, which is encumbered by law etc., or something to that effect. I leave it to someone who knows about this encryption to decide exactly what it should say. QuickFox 10:18, 18 January 2007 (UTC)

It's correct as it is, and I don't know why you think it's strange. See the illegal prime article for more information. --Zundark 13:47, 18 January 2007 (UTC)

Oops! My bad, I should have checked that link. Reading that it makes perfect sense. I've modified the description a little so the meaning becomes clearer. QuickFox 22:50, 19 January 2007 (UTC)

Euclid's proof

Ok, Superdeterminism has been repeatedly inserting his claim that Euclid's proof is not a reductio ad absurdum, but a direct proof. Superdeterminism is making a mistake that many people make when they read Euclid's proof, and when I asked him for a citation, he said (in an edit summary):

MATHEMATICS AND ITS HISTORY , John Stillwell, 1989, page 29

Can Superdeterminism or somebody else please verify what's actually said on page 29 of that book? Meanwhile I'm going to remove the error from the article and reinstate the correct version. -GTBacchus^(talk) 18:38, 10 April 2007 (UTC)

To decide this simple question one would need to read the formulation of the theorem and the proof by Euclid, word by word. In principle, not Euclid's proof but simply a proof should be direct when the theorem is formulated as:

Theorem  For every finite set A  of primes there exists a prime which does not belong to A.

I prefer this form to be stated first, and the standard form to be stated only afterward:

Theorem  The set of all primes is infinite.

-- Wlod (talk) 09:37, 23 November 2007 (UTC)

1989 is the first edition, but google books has the relevant page of the second edition of the book. It's page 39, See here. - Ehheh 18:46, 10 April 2007 (UTC)

Superdeterminism is Archimedes Plutonium, a well-known crackpot. —David Eppstein 18:57, 10 April 2007 (UTC)

With an IP address he added this and more to the article last year (diff):

"No-one in the history of mathematics has given a valid proof of the Infinitude of Primes, reductio ad absurdum method, until Archimedes Plutonium circa 1992".

During 14 years he has made a large number of Usenet posts [3][4] claiming that almost everybody else make false proofs of the infinitude of primes. Many mathematicians have patiently argued with him without result. I don't think anybody has ever supported his claims. They clearly violate Wikipedia:Original research and don't belong in Wikipedia articles. PrimeHunter 19:35, 10 April 2007 (UTC)

Oh, I think they may belong in Wikipedia. In the Archimedes Plutonium article. He is, after all, not just a crackpot but a well-known crackpot. I agree that they don't belong in serious mathematics articles, though. —David Eppstein 20:32, 10 April 2007 (UTC)

Archimedes Plutonium aka Superdeterminism continues to vandalize the wikipedia page on prime numbers. His contributions are unmathematical and ungrammatical. This part of the wikipedia page is fairly elementary and primarily of historic interest. Superdeterminism seems to have no expertise in the history of science and never quotes primary or secondary sources. Instead he expresses an opinion incoherently which seems not to have been accepted in any verifiable source. This seems contrary to wikipedia policy. I hope that he is blocked from editing if he continues this crackpot vandalism. Mathsci 06:23, 12 April 2007 (UTC)

1 , prime or not?

If a prime has only two divisors/factors, then 1 is not prime. It is by definition indivisible/integral, therefore has no divisors. The debate and confusion concerning primes, seems to originate at the inital definitions, where 1 (the unit), is included in the set of multiples (2,3,...). These are two different classes of objects. Some multiples can be partitioned into uniform subsets (composite). The multiple that can't are primes. The integer 1 cannot be partitioned.02:39, 14 April 2007 (UTC)phyti

By the definition in this article, 1 is not a prime, because it only has one factor. Defining it as a prime can also work, with appropriate changes to a few theorems. Both definitions are sometimes used; just need to pick one and be consistent. Dicklyon 02:45, 14 April 2007 (UTC)

1 is excluded from being prime to allow the unique prime factorization theorem to be true. If 1 were prime then an arbitrary count of 1's could be included in any prime factorization which would disprove the uniqueness of prime factorization. This is important for arithmetic, but it is really only a technical distinction. Robomojo (talk) 06:57, 2 March 2008 (UTC)

I don't see the problem. We could, when any number is given, add an arbitrary count of 0's after a decimal point or to the beginning of the number, 0003.0000 = 3. If 1's were included in a factorization, sure, a person could add an arbitrary count of 1's, but why would a person do that? Banaticus (talk) 07:54, 3 August 2008 (UTC)

The fundamental theorem of the arithmetic is flawed. If one is not a prime number then it cannot be used as a factor in other numbers, such as "2" and "3"... You could eliminate those as well, but this starts a chain reaction eliminating all prime numbers. —Preceding unsigned comment added by 209.193.46.76 (talk) 18:03, 8 September 2008 (UTC)

Excluding 1 from the list of prime numbers is purely "a matter of convenience" (Primefan) and it "does not express some deep fact about numbers: it just happens to be a useful convention" (Gowers). If you think 1 is a prime number and you compile a factorization table, you might write something like

1	1^x
2	1^x * 2
3	1^x * 3
4	1^x * 2^2
5	1^x * 5
6	1^x * 2 * 3
7	1^x * 7
8	1^x * 2^3
9	1^x * 3^2
10	1^x * 2 * 5

Since you're gonna write that "1^x *" every single time, why not just leave it out and let the reader see it? But then some smart-ass shows up and says you forgot the 1s. So in response you change the definition and save ink. Anton Mravcek (talk) 20:02, 8 September 2008 (UTC)

Here's another reason it's more convenient to take 1 to be a unit, not a prime.

1 a prime → (1) a prime ideal → Z/(1) an integral domain → The zero ring a subring of a field → The one element ring is a field → The empty set is a group

and somewhere in that chain one would have to have an inconvenient exceptional case. Cleaner not to start down that road. Richard Pinch (talk) 21:00, 8 September 2008 (UTC)

still inconsistent

Quote[ Prime number 1.In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. Prime divisors 2.The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of 1 or more primes in a unique way, i.e. unique except for the order. ]

Statement 1 (requiring 2 factors/divisors minimum for all), contradicts statement 2 (allowing 1 factor for some).

No, it does not. Primes have 1 prime factor, and also 1 as a factor. Other numbers have more. Dicklyon 01:55, 16 April 2007 (UTC)

If 1 is a divisor (by statement 1), then it is a factor. If it is a factor and not prime, then statement 2 is false.

Yes, 1 is a divisor, but is not a prime, and therefore not a prime factor. If it was considered a prime, then factorization into primes would not be unique, because you could use any number of factors of 1. Dicklyon 01:55, 16 April 2007 (UTC)

Within the context of elementary math (+,-,*,/), attempt to rescue the FTA definition!64.24.177.121 01:21, 16 April 2007 (UTC)phyti

There is no problem. Read it again more carefully. Dicklyon 01:55, 16 April 2007 (UTC)

one?

Can the article please include a sentence or paragraph explaining why 1 is or isn't a prime?--Sonjaaa 23:27, 8 May 2007 (UTC)

See the 6th paragraph of Prime number#History of prime numbers, starting

Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labelling 1 a prime,

— Arthur Rubin | (talk) 00:04, 9 May 2007 (UTC)

I think that this makes more sense

Shouldn't 4, 6, 8, and other even numbers whose only factor other than themselves and 1 is 2 be considered prime numbers? After all, it's a little unfair if 2 is the only even prime number. — Preceding unsigned comment added by 124.180.235.178 (talk)

Wikipedia does not make up definitions. We base our content on what has already been published by reliable sources, and prime numbers is an old well-defined concept. PrimeHunter 02:07, 20 May 2007 (UTC)

By the way, numbers of the form two times a prime are even semiprimes (sequence A100484 in the OEIS). PrimeHunter 02:14, 20 May 2007 (UTC)

Yes, 2 is the only even prime, a multiple of 2. And 3 is the only prime that's a multiple of 3, and 5 is the only prime that's a multiple of 5, ... Do you see a pattern here? — Loadmaster 18:27, 13 June 2007 (UTC)

Formulas yielding prime numbers

There are, there are... say what they are. Give the names. I know there is a subarticle, but at least giving a few names wouldn't hurt, even if you don't expand them here. The use of "there are" detracts from the credibility of the article, which is a shame because as far as I can see it's a good article. Shinobu 03:36, 27 May 2007 (UTC)

I don't see your point. Do you mean names of fomulas, or names of people connected with those formulas? How would this help? If the names you want are in the sub-article, perhaps you could move them to this article yourself? Cheers, Doctormatt 05:34, 27 May 2007 (UTC)

There are some formulas elaborated by Lainé Jean Lhermite Junior

Everyone can elaborate similar formulas with the understanding of a famous model!

$P_{n}=\sum _{i=1}^{2^{n}}\left(\left\lfloor {\frac {1+\sum _{m=1}^{i}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor }{\frac {\left(m!\right)^{2}}{m^{3}}}}\right\rfloor \right)}}{n+1}}\right\rfloor \times {\left\lfloor {\frac {n+1}{1+\sum _{m=1}^{i}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{2}}{m^{3}}}\right\rfloor }{\frac {\left(m!\right)^{2}}{m^{3}}}}\right\rfloor \right)}}}\right\rfloor }\times {i}\times {\left({1-\left\lfloor {\frac {\left\lfloor ({\frac {\left(i!\right)^{2}}{i^{3}}}\right\rfloor }{\frac {\left(i!\right)^{2}}{i^{3}}}}\right\rfloor }\right)}\right)$

$P_{n}=\sum _{i=1}^{2^{n}}\left(\left\lfloor {\frac {1+\sum _{m=1}^{i}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{6}}{m^{7}}}\right\rfloor }{\frac {\left(m!\right)^{6}}{m^{7}}}}\right\rfloor \right)}}{n+1}}\right\rfloor \times {\left\lfloor {\frac {n+1}{1+\sum _{m=1}^{i}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{6}}{m^{7}}}\right\rfloor }{\frac {\left(m!\right)^{6}}{m^{7}}}}\right\rfloor \right)}}}\right\rfloor }\times {i}\times {\left({1-\left\lfloor {\frac {\left\lfloor ({\frac {\left(i!\right)^{6}}{i^{7}}}\right\rfloor }{\frac {\left(i!\right)^{6}}{i^{7}}}}\right\rfloor }\right)}\right)$

$P_{n}=\sum _{i=1}^{2^{n}}\left(\left\lfloor {\frac {1+\sum _{m=1}^{i}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{4}}{m^{5}}}\right\rfloor }{\frac {\left(m!\right)^{4}}{m^{5}}}}\right\rfloor \right)}}{n+1}}\right\rfloor \times {\left\lfloor {\frac {n+1}{1+\sum _{m=1}^{i}{\left(1-\left\lfloor {\frac {\left\lfloor {\frac {\left(m!\right)^{4}}{m^{5}}}\right\rfloor }{\frac {\left(m!\right)^{4}}{m^{5}}}}\right\rfloor \right)}}}\right\rfloor }\times {i}\times {\left({1-\left\lfloor {\frac {\left\lfloor ({\frac {\left(i!\right)^{4}}{i^{5}}}\right\rfloor }{\frac {\left(i!\right)^{4}}{i^{5}}}}\right\rfloor }\right)}\right)$

Everyone can chose a number as a power!

We shall indicate another method later .

Please sign your posts. When can we expect to see your other method? Will it also look like this Haitian hoax [[5]] ? 84.210.139.189 14:35, 25 August 2007 (UTC)

ROHA?

Anomymous editors have twice recently begun their edit summaries with ROHA. Anyone know what the heck that means? Thanks. Doctormatt 03:13, 19 July 2007 (UTC)

It's an editor who has a changing IP and doesn't want to register but uses the edit summary ROHA as a type of identification instead of an editor name. Of course, other IP's could choose to say ROHA, so it's not confirmed that all ROHA edits are by the same. I have recommended registration to ROHA who said no. PrimeHunter 11:35, 19 July 2007 (UTC)

Yes, they all seem to be the same - as per this comment by him and many others all over the encyclopedia. I don't know what his editing here has been like, but this might be relevant - my experience has been that he tendentiously edits, insisting on his POV changes, until you finally semi-protect the article. And he always comes back. Tvoz |talk 07:31, 9 October 2007 (UTC)

Ok, guess I wrote in the wrong spot. Sorry.

Ok, the resource thing is, um, well... debatable. (What does Wiki define as a "Reliable Resource"? Plagiarized information?)

Any-way, as stated, that entry was my own, and as far as it matters, I am no-one, as are the other theorists mentioned. (Great, they theorized incorrectly, and because the PRIME NUMBER 1 breaks their formula... "Lets change the laws of the rules, so our formula looks right!" No, it is the formulas that are flawed by the poor representation of numerical order. You have to subtract 1 from every number to see real numbers. Two is technically NOT a prime in reality, it is a prime by the representation we use for numbers.)

1,2,3,5,7,11,13,17 would be... 0,1,2,4,6,10,12,16 ...

Notice that 1 = 0, as a division value. Any object divided one time is 2 halves. Any object divided zero times, is one whole object. You cut one division in a hot-dog, you have two halves. That hot-dog was divided once, divided by one slice. The portions are two, there are two equal portions. Don't let bad child-math mess with your head. You make formulas off bad child-math, and you get bad formulas.

As for my edit, possibly, I could have placed it in a better location. (This is my first time using this Wiki.)

There are infinitely many prime numbers The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Consider any finite set of primes. Multiply all of them together and add one (see Euclid number). The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one.

The statement above, has the most horrible contradiction, and has been proven false. It is a theory, that is depreciated, and no longer relevant. If that statement/theory were true, we could easily determine all primes, by multiplying and adding one to any set of lower primes. However, it fails to take into account, that it only works on ALL LOWER PRIMES, known at that time. The theory holds false, in the second half of primes. He has proven nothing, except by a few samples of low prime numbers. The only truth to his theory, is that adding 1 to any product of two primes, will give you an ODD number, because the product is always EVEN, but it is NOT always +1 away from a prime. (He just got lucky, and used formulated values to "estimate" the numbers were possibly prime, not prove they were prime.)

http://www.ieeta.pt/~tos/gaps.html

You can clearly see, the progression of gaps, rises with each new set of primary multiplication tables created by these prime numbers. 2 has erased half the numbers chances, 3 has erased 1/3 of the numbers, but overlaps 2 by 1/6, 4 is part of 2's multiplication table, 5 has erased 1/5 of the possibility, overlapping 2 tables and 3 tables... Each prime number continues to erase the possibility of infinity... Thus, by that same deduction, there is an ending to prime numbers, a point in which all lower primes, not all at once, are a multiple of the numbers that follow that last prime.

Remember, numbers are fictional, nor real. We invented numbers in relation to objects. Just as we invented the second dimension of numbers, called decimals, to represent the infinite space on the second plain, space between whole numbers. If infinity lies between each number, then no numbers actually exist. 2 is infinitely away from 1. Until you grasp that concept, and realize that "PRIME NUMBERS" are a "Man made classification of numbers with a unique attribute.", just as "EVEN" and "ODD" and "REAL", and "WHOLE", and "DECIMAL"... then, and only then, will you be in my world.

The, "Child explanation.", of division, is maths biggest failure. You can not divide by 0 because of this... You can not divide by 1. Not a real possibility. Any object divided ONCE = 2 parts. Any object divided ZERO times = 1 part.

Division is a "Child's" game. We invented it to show them a way to understand division. We messed-up by saying, "Divide by", and calling 2 a division factor. The factor is 1, which results in two division containers of equal proportions. 2 is the quantity of proportions, not the number of times it is divided.

It just happened to fit well with the inverse of multiplication. (3*2) = (3/2) Divided by two??? No, divided IN two! You have one division, which results in 2 portions. Take a hot-dog, divide it once... how many pieces do you have? See, 1/1 = 2.

Thus, to understand primes... subtract 1 from every number. (Then you will understand them, and see the relation.)

Relation is simple... take a clear piece of tape, write the factors of two but not two itself, in the numeric order, spaced properly...
Do the same for 3, not writing 3...
and 4, 5, 6, 7, 8....

When you are finished, overlap the tape. If you see a number, it is not prime. If you do not, and the number is below the last piece of tape you wrote factors for... the missing numbers below that number, are all prime. Eventually, if you continued, you would overlap many numbers, and fill the missing numbers, to infinity. Ultimately, there will be a point where there are no more missing numbers.

Like I said, in the post which was removed...
Prime numbers can NOT be infinite because... (By the rules of PRIME NUMBERS.)
(Infinity/2) = Removal of all even numbers, including negative and decimal.
((Infinity/2)-(non-overlapping multiples of 3)) Multiples that do not overlap "EVEN" numbers.
((Infinity/2)-(NO3 and NO5 and NO7 and NO11...)) Non-overlapping multiples grows exponentially, since each prime has infinite multiples. Eventually, there is no more non-overlapping multiples. Proven by the expanding distance between prime numbers, and the reduction of available primes per (Length of number).

If you take 1,000 sets of numbers, group 1 may have 300 primes, group 2 may only have 280 primes, group three may only have 200. (The only truths about primes, is that they must get larger, and farther apart, in order to exist. This draws one big ugly arch when plotted. There is a point in that plot, where the arch intersects 0. That is the "Last Prime Number".

The other thing I had said, and proven, was that you don't have to do one single division to find any prime numbers. If you know the prime number, and the (N)N, of that prime, incrementing X in (N)X, leads you to your next limit range. You simply increment X when the number you are testing, matches one of the prime numbers values of (N)X.

[1]* (Forget about it, that is the base of your formula, it is a factor in all whole numbers.)
[2]* (Forget about it, this is the first "Factual" number, it HAS to be prime.)
[3]* (3*3) = {9}, thus, nine is not prime, when you get to it, you increment (3)X from (3)2 to (3)4 but not now.
|4|- This is assumed not-prime, through formula, or you can waste time tracking even values like you do for (3).
[5]* (5*5) power = {25}, save that for later. This is a prime, because it is not in the list of NOT-PRIME, which contains EVEN and {9}, and now also {25}.
|6|- This is even, throw away...
[7]* (7*7) power = {49}, This is PRIME, it is not EVEN {9} or {25}, You can top checking at {9} because half of 7 is int{3} anything higher than 3 will result in a remainder if it was divided.
|8|- EVEN, throw out...
{9}- Odd, but it is part of 3's multiples. Here is where you increment (3*3) = {9}, to (3*5) ={15}, Ok, this is not even.
|10|- EVEN Throw out...
[11]* (11*11) = {121}, PRIME, not in list, ({15}, {25}, {49}) List stops at 15, because (11/2) < 15...
|12|- EVEN Throw out...
[13]* (13*13) = {169}, PRIME, not in list, ({15}, {25}, {49}, {121}), Notice, you are not checking or changing the values of the higher primes, and this data can be calculated as needed, instead of wasting time doing it now. You also only store that ONE LAST number... the priors are irrelevant, as they determined your lower numbers already checked.

Not one division, only simple and fast multiplication, which could easily be addition... (3 + (3) + (3)) or (3 + (3)x), with x being x = x + 2, on an increment. Again, if you have space, (Y + (6)), {6 is 3 + 3) Y started as 3, and goes 3, 9, 15, on each 6 increment.

The list of "Checking Factors" stays exponentially small, as each prime gets larger, and gaps get wider.
(104729 * 104729) = 10968163441 (That will last, unscanned until you get to)... (10968163441 * 2) = 21936326882 —The preceding unsigned comment was added by 75.92.77.172 (talk)

Your theories have no source that satisfies Wikipedia:Reliable sources, but there are many reliable sources that contradict them. The work violates Wikipedia:No original research and Wikipedia:Verifiability, so it should not be added to any article. Wikipedia is not the place to introduce your own research. You can look for another website that allows it, or make your own. PrimeHunter 14:24, 26 July 2007 (UTC)

Excuse me for asking you here, but I think you can tell me whether this is a correct understanding of what you have posted so far: 1) Prime numbers are particular integers that are distinguished by their non-divisibility by other integers without getting a fractional remainder, all the time excluding the undistinguished property of divisibility by 1 or the number itself. 2)Your definition of division by a number "n" is that it means cutting the whole n times to leave n+1 pieces. 3) You don't like the school-taught definition of division by a number "n" which is that it means cutting the whole into n pieces. Is this correct so far? Cuddlyable3 15:21, 25 August 2007 (UTC)

If Euclid's statement is prefixed "Consider any set of primes that is complete up to the highest known prime." Do you agree that one can always multiply all of them together and add one, though not necessarily an easy task with big numbers? Euclid does not say the resultant big number IS a prime although it might be. It might not be a prime. If it is not a prime then can its factors all be found in the known set? Cuddlyable3 15:21, 25 August 2007 (UTC)

This has to be one of the most hilarious things I've ever read. Are you sure your not the guy from timecube? —Preceding unsigned comment added by 65.118.202.4 (talk)

This explanation is incoherent and blatantly wrong in many places, but appears to have some relationship to the Sieve of Eratosthenes, which is in fact quite an efficient method of finding the first k primes without divisions. See Prime_number#Finding_prime_numbers. Dcoetzee 18:48, 14 January 2009 (UTC)

The simplest proof

"The simplest proof of the infinitude of the prime numbers is the 1730 proof given by Christian Goldbach." states the article and goes on to talk of Fermat numbers. I dunno what exactly we mean by simple here but this proof struck me as a damn sight more complex than the one by Euclid given earlier in the section. Jɪmp 07:12, 29 July 2007 (UTC)

I agree. It should be removed or rewritten with a source. It was added [6] by an editor who makes a lot of unsourced and frequently false additions to articles about prime numbers. PrimeHunter 12:21, 29 July 2007 (UTC)

See the page on Fermat number to see why I believe that Goldbach's proof is much simpler than Euclid's proof. PhiEaglesfan712 19:02, 29 July 2007 (UTC)

I disagree, but it doesn't matter much what we think. It matters what reliable sources think. Please don't insert your own opinions in articles. Do you have sources that call this the simplest proof? PrimeHunter 21:15, 29 July 2007 (UTC)

I added the original source letter in Latin from Goldbach to Euler, dated July 1730, from the Dartmouth archive of Euler's correspondence. The history is described on page 8 of the survey book by Narkiewicz that I added in the references. It seems inappropriate to add the details of Goldbach's proof, when no other proof than Euclid's is described in detail. However, adding the proof in the article on Fermat numbers with a link from the sentence about Goldbach's proof - insert the word "proof" with a link - seems a completely reasonable solution, which I hope would satisfy everyone. Mathsci 22:39, 29 July 2007 (UTC)

The proof was already there, so I added the link to the relevant subsection of Fermat numbers myself. I hope this is OK. Mathsci 01:25, 30 July 2007 (UTC)

It looks good to me. PrimeHunter 01:38, 30 July 2007 (UTC)

It definitely looks a lot better and more professional than my original edit on July 11, 2007. PhiEaglesfan712 16:24, 31 July 2007 (UTC)

Euclid's proof is simpler

Goldbach's proof is interesting, even briliant, while Euclid's is simpler, more natural, more elegant. Actually, the two proofs can be viewed as nearly identical, except, that Euclid's (after a minor variation) uses a simpler, natural, direct sequence. One may say, that Fermat's sequence is more elegant than analogous Euclid's sequence. But Euclid's proof is more elegant and simpler, see:

http://wlod.blogspot.com/2007/07/step-beyond-euclid-and-fermat-1.html

BTW, years ago, I discovered a proof which uses Mersenne's numbers, and have posted it all over Internet. It has survived, I believe, on sci.math.research, around 1995 (there were earlier, pre sci.math.research posts, which didn't survive, and there are later posts on pl.sci.matematyka, where I have considered some further variations). -- Wlod 20:38, 10 November 2007 (UTC)

Please note that this is not a forum: it is a talk page for discussing edits to this article. --Mathsci 04:24, 11 November 2007 (UTC)

FTA in intro

Shouldn't a quick mention of the FTA be in the introduction near the beginning, as a sort of motive for studying primes? —Preceding unsigned comment added by AbcXyz (talk • contribs) 19:32, August 26, 2007 (UTC)

What is FTA? -- Wlod 20:42, 10 November 2007 (UTC)

Fundamental theorem of arithmetic. PrimeHunter 20:53, 10 November 2007 (UTC)

Yes, I also think that FTA should be stated early, and right away for (positive?) rational numbers. Then instantly one gets a bunch of irrational numbers, like  $(2^{3}\cdot 3^{-2})^{\frac {1}{n}}$  for arbitrary integer  n,  such that  |n| > 1. -- Wlod 08:12, 11 November 2007 (UTC)

Finding Prime number in COBOL (Shortest Code)

I wrote shortest cobol code for finding Prime numbers within a limit.

       IDENTIFICATION DIVISION.
       ENVIRONMENT DIVISION.
       DATA DIVISION.
       WORKING-STORAGE SECTION.
       77 A PIC 999.
       77 LT PIC 9(5).
       77 B PIC 9(5).
       77 C PIC 9(5).
       77 D PIC 9(5).
       01 AAA.
       02 AR PIC 999 OCCURS 999 TIMES.
       PROCEDURE DIVISION.
       
       PARA1.
                DISPLAY "ENTER A LIMIT".
                ACCEPT LT.
                PERFORM VARYING A FROM 1 BY 1 UNTIL A > LT
                PERFORM VARYING B FROM 1 BY 1 UNTIL B > A
                DIVIDE A BY B GIVING C REMAINDER D,
                IF D=0
                COMPUTE AR(A) = AR(A) + 1.
                PERFORM VARYING A FROM 1 BY 1 UNTIL A > LT
                IF AR(A) < 3
                DISPLAY A.
                STOP RUN.

can I add above code in Prime number article? --Karthixinbox

Unfortunately not, because WP has a policy WP:NOR against the inclusion of original research in articles. Also please leave your comments at the end of the discussion page rather than the beginning. (I moved your contribution here.) --Mathsci 15:50, 2 September 2007 (UTC)

Also note that many people have written much shorter and faster programs in languages better suited for mathematics, and Wikipedia is not a code repository, so your code would be inappropriate even if it wasn't considered original research. Here is a short slow prime program I wrote in C:

main(){unsigned p=1,d;for(;d=p++;d||printf("%u\n",p))while(p%d--);}

I have also written fast prime programs in C but all are inappropriate for Wikipedia. PrimeHunter 16:25, 2 September 2007 (UTC)

Should we really talk about class n+ and class n- in this article ?

Cf in the article:

Classification of prime numbers ... Two ways of classifying prime numbers, class n+ and class n−

This classification looks too minor for inclusion here. (And its importance is not explained.) Perhaps we should add it in the "see also" section instead. --FvdP 17:51, 26 September 2007 (UTC)

I agree it should go. If there is no other article about it then it can just be removed completely. PrimeHunter 19:55, 26 September 2007 (UTC)

The Erdos-Selfridge classification should at least be a see-also. Both PlanetMath and Mathworld have articles on this very topic. But surely the E-S is not the only way there is of classifying primes? Surely there are others? Anton Mravcek 00:07, 27 September 2007 (UTC)

There are many ways. If we have another article mentioning this one then a link to it is fine, but I don't know another article. PrimeHunter 00:38, 27 September 2007 (UTC)

If it comes down to it, I could simply copy the PlanetMath article about the Erdos-Selfridge (it's GFDL). Anton Mravcek 22:10, 27 September 2007 (UTC)

Euclid proof

The Euclid proof contains claim: "The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one."

Why is it justified to claim that the remainder is one? I fail to see it as obvious (but maybe it's just me). As formulated as a nice little problem: Prove that remainder of an Euclid number, when divided with it's primorial factors, is always one."

I did find this proof more understandable: http://mathworld.wolfram.com/EuclidsTheorems.html. A bit longer but avoids the remainder claim. --Ketorin 19:43, 1 October 2007 (UTC)

Uh, got it. Of course, it's simple. If $P=p_{1}p_{2}...p_{N}+1$, for each prime $p_{i}$ we can rewrite: $P=p_{i}(p_{1}p_{2}...p_{(}i-1)p_{(}i+1)...p_{N})+1$. Mark $p_{1}p_{2}...p_{(}i-1)p_{(}i+1)...p_{N}=K$, thus we can rewrite $P=p_{i}K+1$, which clearly has a remainder of one. --Ketorin 20:53, 1 October 2007 (UTC)

This is NOT the end of the story. How do you know that having remaider 1 prevents divisibility? Instead, you need to observe that a natural number k > 1 cannot divide two consecutive integers n and n+1 (because it would divide their difference = 1, which is a contradiction). Now indeed, this is the end of the story. -- Wlod 20:48, 10 November 2007 (UTC)

Removed sentence about generalizing Euclid's proof to any infinite UFD

There was a sentence claiming that Euclid's proof "easily generalizes" to any infinite UFD. First of all it is not even true that an infinite UFD must have infinitely many prime elements, or any prime elements at all: consider the case of any infinite field. (Perhaps a field is a trivial sort of example of a UFD, but it certainly is a UFD according to the universally agreed upon definition.) Second, in an arbitrary UFD a more interesting question than whether the prime elements are infinite in number is whether there are infinitely many pairwise nonassociate prime elements, or equivalently whether there are infinitely many principal prime ideals. In any discrete valuation ring -- e.g. Z_{(p)},the integers localized at the prime ideal (p) -- there is only one prime element up to units. Moreover although Z_{(p)} contains infinitely many prime elements (all associate to each other) Euclid's proof does not show this: you start with p, you add 1, and you get p+1, which is a unit.

In summary, the sentence is wrong in many ways, so I deleted it. Insert cranky mathematician's complaint about getting things wrong in important articles here. Plclark 09:09, 8 October 2007 (UTC)Plclark

Another dubious sentence

"In both of the two above examples, the fundamental theorem of arithmetic (Every natural number can be ‘uniquely’ decomposed into a product of primes) does not apply."

What is this sentence supposed to mean? Obviously the fundamental theorem of arithmetic in the form it is quoted here, being a theorem about natural numbers, cannot literally "apply" to an arbitrary commutative ring or to a knot. If the sentence is an allusion to the existence of rings of algebraic integers in which unique factorization fails, it should be made less obscure, and also an example should be given -- the only two examples of more general rings of integers cited in the article do in fact have unique factorization (in the sense in which the term is applied to general integral domains). Moreover, isn't it the case that knots DO factor uniquely into prime knots? Please explain what's going on here or this sentence will be deleted too. Plclark 09:26, 8 October 2007 (UTC)Plclark

No one has come forth to explain it, so I removed it. 74.224.198.170 03:55, 12 October 2007 (UTC)Plclark

Other minor fixes

I noticed a few other minor things:

1) The last property of primes does not appear to be of any significance; why are the congruence classes for primes modulo 6 and 30 given rather than modulo some other number N?

2) A reference to Kummer's proof of the infinitude of primes is needed.

3) A supersingular prime is not a prime given by a special formula. In fact, it doesn't make sense to say whether a prime number by itself is supersingular; the question is relative to an elliptic curve over a number field.

4) The use of places in the valuation-theoretic sense is more widely used than just in class field theory. Better would be "In valuation theory...."

5) "Polignac's conjecture" should, it seems, be "de Polignac's conjecture."

6) Isn't "Many believe" a construction than wikipedia frowns upon? Who believes, for instance, that there are infinitely many Fibonacci primes? (For instance, I have no strong feelings either way.) Plclark 09:54, 8 October 2007 (UTC)Plclark

On your last point, the many who believe are professional mathematicians who have been unable to construct a proof or disproof of this, and aficionados who find the empirical evidence convincing. But to keep things simple, I've reworded "Many believe" to "It is conjectured that" Anton Mravcek 23:25, 8 October 2007 (UTC)

Not being able to construct a proof or disproof is not very good grounds for belief. (Note that one does not have to believe either that "X is true" or "X is false" even though one knows that at least one of the two is true.) Moreover, a finite amount of evidence may be suggestive that a set is infinite, but "convincing" is a little much.

None of that is really the point though. To say "it is conjectured that" without saying by whom is a little better than "many believe", since to verify "it is conjectured that" one could find a written document which contains such a conjecture, whereas it is not clear what would constitute a verification of "many believe." But better still would be to say who has conjectured it and provide a link: that way, evidence for the conjecture can be evaluated. In the absence of any interesting evidence or heuristics, I think that the statement

"It is unknown whether there are infinitely many Fibonacci primes."

is simpler and more honest. Similar remarks apply to some of the other conjectures; e.g. how can one verify the statement "It is widely believed that there are infinitely many Mersenne primes"? In this case I do know of some specific leading mathematicians who believe this, although I myself do not (nor do I disbelieve it). I suppose I could round up a lot of other mathematicians who feel similarly, but how many would I have to find to contest the "It is widely believed that"? Better I think is to state the problem and cite at least one reputable source that makes the conjecture.

Heuristic estimates are in favour of infinitely many Fibonacci primes, based on the prime number theorem compared to the growth rate of Fibonacci numbers with prime index and their apparent lack of systematic divisors, but I don't have a reliable source for this argument. http://primes.utm.edu/glossary/page.php?sort=FibonacciPrime says: "It seems likely that there are infinitely many Fibonacci primes, but this has yet to be proven." I would be surprised to see a serious mathematician who guessed there are finitely many Fibonacci primes.

Regarding your 1) Congruence classes modulo the smallest primorials 2, 6, 30 are of special interest. They are for example used to speed up many prime sieves. Going from one congruence class to another can only reduce the possible prime values if the second class has a prime factor which is not in the first.

Regarding your 3) Supersingular prime has two different definitions and both are sourced in http://mathworld.wolfram.com/SupersingularPrime.html. One of the definitions gives a specific list of primes.

Regarding your 5) "Polignac's conjecture" and "de Polignac's conjecture" are both used by reliable sources [7]. Wikipedia should choose one and be consistent. If you want to change the choice then a discussion should be at Talk:Polignac's conjecture and not here. PrimeHunter 15:37, 12 October 2007 (UTC)

Congruences classes mod 6 or 30 may be of special interest for the division-trial algorithm to factorize numbers, but they are of no special interest to primes in general, I would say, so they shouldn't stay here. --FvdP 17:17, 12 October 2007 (UTC)

For the Fibonacci primes, I do quite agree with Plclark, "widely believed" is bad (even though *I* happen to believe it), "it is unknown whether" is better, and your heuristic argument is also better than just "widely believed". --FvdP 17:20, 12 October 2007 (UTC)

Regarding 1): your explanation about primorials sounds reasonable. Of course this explanatory information needs to be in the article itself. Will you put it in?

Regarding 3): I am intimately familiar with supersingular primes based on the second definition, by which I mean I have read many papers (and written a few) on supersingularity in the latter sense. I have not seen a single usage from a primary source (mathworld is a proto-wikipedia; it is not a primary source) for "supersingular" in the first sense. Of course I have not read every math paper, but in an attempt to confirm my suspicions, I searched for "supersingular prime" on MathSciNet and can confirm that 37 out of the 37 references are to supersingular in the second sense. I therefore contest that mathematicians do not in fact use supersingular in the first sense, and I have said as much on the page. Do you have any verification that they do use it in this way?

Regarding 5): Since Wikipedia identifies the mathematician as "Alphonse de Polignac", it seems to simply be an error coming from unfamiliarity with French names to say "Polignac's conjecture." Most anglophone writers who are experts on prime numbers are not particularly expert in French, so I don't think the appearance of "Polignac's conjecture" in the literature should be taken as serious evidence of a preference versus "de Polignac." You're right that the name should be changed on the other page -- I don't know exactly what's involved in doing this. Plclark 18:56, 12 October 2007 (UTC)Plclark

I changed "It is conjectured that...Fibonacci primes." to "It is unknown whether there are infinitely many Fibonacci primes." I agree that it would be better to say that there are some heuristic arguments supporting the infinitude, but this should be done in a useful and verifiable way: i.e., with a reference and preferably a link. Isn't this the wikipedic way? Plclark 19:01, 12 October 2007 (UTC)Plclark

I think some well-known (i.e. with Wikipedia articles) number theorists have conjectured that there are an infinte number of Fibonacci primes. It would be better just to {{fact}}-tag it and see what develops. — Arthur Rubin | (talk) 19:03, 12 October 2007 (UTC)

First, the idea that a well-known number theorist means one with a wikipedia article is terribly wrong. Most well-known number theorists do not have wikipeida articles, and there are many wikipedia articles about number theorists that are not otherwise very well-known. This is not the point here, but you should know better than to say something like this.

Second, "you think" that someone has made a conjecture and your reaction to this is not to find a citation but to change an incontestable neutral statement to a statement which is difficult to verify? Whose job is it to find the citation if not the person who makes the claim in the first place?

For what it's worth, I too think that lots of people, including many expert number theorists, believe that there are infinitely many Fibonacci primes. (And I have said before, I don't disbelieve it.) This does not release anyone from the obligation of tracking down the citation. Since at this point it clearly seems easier to add the citation myself than get several wikipedians to recognize the shoddiness of their scholarship, I have done several searches to see if I could find a reputable source that makes this conjecture. But I couldn't find anything, and I think it's worth reflecting why many of the people who write papers finding more Fibonacci primes do not include in their paper the conjecture that there are infinitely many. (Hint: it's not because they think there are only finitely many!)

I'm not going to change the sentence back, but I am going to warn you all that I think you are being bad scholars. Plclark 22:29, 14 October 2007 (UTC)Plclark

I have added [8] a reference to [9]. I guess the conjecture is mentioned in at least one of the 3 references there but I don't know. Note that most people who find special prime records and large primes don't write papers about it. They just announce it on the Internet and/or submit to a record list. I have set over 100 prime records and written a single math paper (which was about pseudoprimes and didn't mention prime records). PrimeHunter 00:17, 15 October 2007 (UTC)

Sigh -- "guessing" at references is again bad scholarship. I have verified that the conjecture does not appear in BMS1988 or DK99. To check Brillhart1999 I would have to physically go to the stacks and look it up. I will do so if you are actually claiming that Brillhart1999 makes the conjecture, but not if you are just lazily hoping that it will.

What do the publication habits of people who search for prime number records have to do with this issue? On the one hand, no amount of record-setting addresses a conjecture on infinitude of a set, and on the other hand, the fact that a group of people who believe something tend not to write down their beliefs certainly does not absolve wikipedists of their standards for verifiability.

There are reasons why mathematicians do not write down everything that they think might be true, and that even when they do they say what they believe they do not call it a "conjecture." Let me ask you this: what additional information is conveyed by saying "It is conjectured that" versus "It is unknown whether"? If there is specific information connoted to you by the former, incorporate that explicitly into the article. Otherwise, why are you so insistent on including vague statements about what certain people believe? Can you not provide the reader with enough information to make up his/her own mind? Plclark 01:41, 15 October 2007 (UTC)Plclark

The reference I added to the article is [10] which says: "As with the Fibonacci primes and the Mersenne primes, it is conjectured that there are infinitely many Lucas primes". I think that is a satisfactory reference for Prime number#Open questions which says: "It is conjectured that there are infinitely many Fibonacci primes".

I haven't and wouldn't add my guess about other references to the article but I think it made sense to mention it here in the discussion so somebody with better access like you could consider looking it up.

It was you who wrote: "I think it's worth reflecting why many of the people who write papers finding more Fibonacci primes do not include in their paper the conjecture that there are infinitely many. (Hint: it's not because they think there are only finitely many!)"

My comment about prime searchers was a reply to that. I think the people who found most of the large Fibonacci primes and probable primes don't write Fibonacci papers at all, so they obviously don't write papers with conjectures about infinitude of Fibonacci primes.

Wikipedia doesn't allow original research. If it did then I could make the earlier mentioned heuristic estimate and show that infinitely many Fibonacci primes are expected from the prime number theorem if certain assumptions about "random" prime behaviour of Fibonacci numbers with prime index are made. Most Wikipedia readers don't have mathematical abilitites to make such estimates by themselves but they might like to know what qualified people think about the possibility of infinitely many Fibonacci primes, so I think it makes sense to just say it's conjectured when the currently examined references don't give more details. On the Internet I have often seen hobby mathematicians speculate about which sets contain infinitely many primes and ask what others think. And I guess relatively few readers of Wikipedia math articles are professional mathematicians. By the way, if a reference with more details is found then the details should probably only be added to Fibonacci prime and not prime number. PrimeHunter 02:36, 15 October 2007 (UTC)

As I said, (not especially good) guessing that certain papers might contain certain references and hoping that other people will do your library work is not good scholarship. It seems clear that you recognized that the reference to the primepages was not such an ideal one: it does not say, We conjecture here that there are infinitely many... It just repeats It is conjectured that. It is also not a reliable reference in the traditional sense; e.g. it does not make clear enough who is making the conjecture and at what time.

As stands, if by "conjecture" one means "Something that some people who are active in the area think", then the given reference does support that. That's good enough for me for now -- I have my own work to do.

However, it would clearly be better if wikipedia mentioned the heuristics which led to this conjecture. Since other mathematicians have doubtless made these heuristics, discussing them is not in reality original research. However, there is still a verification problem. But you can overcome this by either (i) finding a number theory text or paper that gives heuristics of this type, or (ii) writing up such heuristics and putting them elsewhere on the web. For instance, you could write such an article on PlanetMath. Plclark 15:02, 15 October 2007 (UTC)Plclark

Plclark, you seem to be being exceedingly pedantic about a relatively unimportant matter. Conjectures really do not need the kind of justification you are asking for. Often they can be part of the folklore of a subject. Mathsci 17:58, 15 October 2007 (UTC)

An encyclopedia is the place to be excessively pedantic, especially in the face of insufficient pedantry on the part of others. Yes, there are folk conjectures and this may be one of them. However, most mathematicians use the word conjecture very carefully, and are careful to distinguish between a real conjecture and the folklore. You should read the passage in the book by Crandall and Pomerance (on p. 31): it provides information about these heuristic arguments (as well as a precise reference to another paper in which the heuristics are studied quantitatively) and is also very clear that these heuristics are just heuristics, and that one should not necessarily take them too seriously. For the last time, the encyclopedia is not the place to promote very tentative ideas as if they were received mathematical wisdom! I rewrote this passage so as to provide the same amount of information and make clear the provisional nature of these heuristics. I added a a brief allusion to Crandall and Pomerance. If the passage is regarded as acceptable I will format the reference correctly. Plclark 20:42, 15 October 2007 (UTC)Plclark

For what it's worth (and this is WP:OR on my part), the fact that there is a (relatively) fast algorithm for determining whether a Mersenne number is prime, but not for whether a Fibonacci or Lucas number is prime, suggests the reason why fewer people comment on whether there may be an infinite number of them. — Arthur Rubin | (talk) 18:04, 15 October 2007 (UTC)

I have reverted the edits of Plclark because they were very poorly written and verged on the cryptic; I got the impression of a legal document, rather than an attempt to explain fairly accessible mathematics to a general audience, which is what this particular article is all about. The reference to Crandall and Pomerance showed no knowledge of how to add references to a WP article. --Mathsci 01:40, 16 October 2007 (UTC)

Well that's it, then: I'm certainly not going to fight about it. I made many specific changes, eliminating vague sentences like "it seems to have a very high probability of being true" with more precise information. I gave a reference for information about the heuristics about Mersenne and Fermat primes and this was deleted. (In my last post I said that it was not properly formatted, and that if was accepted in content I would go back and format it properly.) I identified four of the problems as Landau's problems in the body of the text. I mentioned that the problem about primes of the form n^2+1 is a special case of Schinzel's hypothesis.

Saying the writing is bad is not a very useful comment. I have written hundreds of pages of mathematics, for researchers and for students of all levels. In the future, I will tell students to not take wikipedia mathematics pages very seriously, and to use something like PlanetMath instead, which is more receptive to critical improvements. Plclark 15:46, 16 October 2007 (UTC)Plclark

I'm afraid saying the the writing is bad is accurate. As for PlanetMath, there are some serious errors due to the "owner" of an article being the only one who believes the content to be accurate, and probably for other reasons. I think Wikipedia, in general, is more accurate than PlanetMath. — Arthur Rubin | (talk) 21:12, 16 October 2007 (UTC)

It is distressing to me that I made and called attention to specific, content-related changes, and these get dismissed on the grounds of bad writing. If it is "bad writing" to insert references to particular conjectures being made and to discuss the relationship of various conjectures to each other, then...well, we disagree too much on what constitutes good writing.

On PlanetMath, the fact that every entry is owned by someone -- and generally someone with substantial mathematical training, most typically a graduate student -- means that there is a specific accountability that is just not present in Wikipedia. Had I found the sentence described in "Another dubious sentence" above on a PlanetMath article, I would have written the author and asked what s/he meant by it, and I would have gotten a response. On wikipedia, when I write in saying that something is unclear and perhaps wrong, most commonly no one steps forward to explain their own work, and others participate only to the point of saying If you think it's unclear, why don't you improve it?

I find it strange that the few professional mathematicians who are involved in wikipedia are not more embarrassed by the extremely uneven quality of the articles. It would seem to be an obvious step to assign some expert to read every article and make sure its content is okay, but obviously this has not been done even on this vital article, since my first activity on this page was to correct a serious mathematical error. That math PhD's like MathSci and Arthur Rubin are willing to revert other people's changes but cannot be bothered to notice and correct errors themselves is discouraging. Plclark 23:11, 16 October 2007 (UTC)Plclark

What you say here is not correct; I don't know why you mention Ph.D.'s. You came to a low-level mathematics article, without knowing how to write mathematics for a general audience. Try taking a look at the large number of more advanced mathematical articles on WP, where even Fields medallists contribute, before making generalisations. What you have written is just wrong. I myself add material bit by bit when I come across yawning gaps. For example at present in real life I am preparing course material on affine Bruhat-Tits buildings, Hadamard spaces and a Property T group of Mumford; some parts of this might get added to relevant WP pages. I think it's safe to say that there is no one expert who could comment on all the mathematics pages. Just take a look at them (assuming that you've had some training in higher mathematics). This particular page is carefully monitored by mathematicians because it occasionally attracts oddballs. On the whole it is written at the correct level. I find PlanetMath articles extremely variable and sometimes misguided, precisely because they are written by graduate students who can lack perspective. --Mathsci 03:15, 18 October 2007 (UTC)

Your claim that I do not know how to write mathematics for a general audience is unwarranted. We disagree seriously on the importance of precision and verifiability in popular mathematics writing. I am certainly not the only one who thinks that it is more important to be accurate and precise in a popular article, because an untrained person is less likely to be able to track down references and spot errors. About expert input: I meant that each article should be read and checked by some expert. I didn't mean that the same expert should read all the articles; that's ridiculous, both because there are too many and because no one is an expert on everything.

You say that this page is carefully monitored by mathematicians, but my point is that this "monitoring" process seems not to include even a cursory reading for mathematical errors, because there was an embarrassing mathematical error on the page. To me, it is of utmost importance to me that the material that appears on a page be correct and that its relevance to the article be clear. For an example of the latter, note that the current page still contains a list of the congruence classes modulo 6 and modulo 30 which contain infinitely many primes, without any clue as to the significance of this. When I brought this up before on this page, PrimeHunter wrote in to say that they were used to speed up prime sieves, but didn't put this explanation into the article. Similarly, anyone from a high school student to the average professional number theorist (e.g. me) is not going to understand the significance of dividing the prime numbers into class n+ and class n-.

Moreover, the page does not contain lots of things about prime numbers which are of vital importance in modern mathematics. Examples: unique factorization of ideals into primes in the ring of integers in a number field, or in an arbitrary Dedekind domain; Frobenius elements in Galois groups of global fields and the Cebotarev density theorem; uses of primes in constructing Euler product representations for L-functions, and on and on. (This is mentioned in passing for the Riemann zeta function, but not sufficiently elaborated on.) Maybe you can add material on some of these topics?

I have read plenty of the other wikipedia articles, including some on advanced topics. In my opinion it would be more useful if you wrote up nice lecture notes and then posted them on the web than if you specifically wrote a wikipedia article on the subject, but of course it's up to you. This dialogue seems to have turned ad hominem and unproductive -- it is really not appropriate for you to tell me what I am unable to do, and conversely I have no right to take you to task for not catching various errors (if you were interested in finding and fixing errors on this page, you would have done so, so clearly your interests lie elsewhere) -- so this will be the end. You can probably figure out that at least one number theorist doesn't like the page on primes very much. What, if anything, to do with that information is up to you. 128.192.134.50 21:00, 19 October 2007 (UTC)Plclark

Why are you telling other editors how to contribute on WP or elsewhere? It is inappropriate and none of your business. This page is for discussing edits to this particular WP article: it is neither a forum nor a pulpit. --Mathsci 16:31, 20 October 2007 (UTC)

Conflict on Goldbach's conjecture

Currently there is a statement: "Every odd integer greater than 5 can be written as a sum of three primes."

However if you follow the link to the Goldbach Conjecture on Wikipedia itself you see that it states: "Every odd integer greater than 7 can be written as a sum of three primes."

However, places like Wolfram state: "Every odd integer greater than 9 can be written as a sum of three primes."

It's easy to prove that this is true for all odd integers greater than 5, but I'm not sure what the actual conjecture stated and Wikipedia needs to be consistent on that.

-- B-Con

There is a very careful discussion of the history of the Goldbach conjecture, including a direct pdf link to Goldbach's original letter in german to Euler, on the WP page on Goldbach's conjecture. I see 5 rather than 7 on that WP page; 7 occurs only when a sum of 3 odd primes is required (i.e. 2 is excluded). Also "greater than" means "strictly greater than" on WP. Do you really know how to prove the weak or ternary Goldbach conjecture? --Mathsci 04:09, 18 October 2007 (UTC)

Ah, my oversight, I failed to note the limitation of "odd primes". Of course I don't claim to know proof of Goldbach's Weak Conjecture, but if it held true for odd integers greater than 7 it would be easy to show it would work for odd integers greater than 5 -- just show that 7 is the sum of three primes, namely 3+2+2. However, the conjecture technically states that every odd number greater than 7 can be expressed as the sum of three odd primes, and 7 is not the sum of any three odd primes. Shouldn't, then, the link and brief description of Goldbach's Weak Conjecture on this page be amended to more precisely reflect the actual conjecture? "...greater than 7 is the sum of three odd primes" is different from "...greater than 5 is the sum of three primes". Why the slight discrepancy? It's not like one is significantly shorter. --B-Con —Preceding unsigned comment added by B-Con (talk • contribs) 07:06, 18 October 2007 (UTC)

If that is what you meant, it is not what you wrote. In mathematics, and also on WP, "greater than" means "strictly greater than"; otherwise we have to write "greater than or equal to". Thus odd and greater than 7 means "≥ 9". (Note that 9 = 3 + 3 + 3.) These later "weak" conjectures are not due to Goldbach, so there's no inconsistency. Cheers, Mathsci 07:49, 18 October 2007 (UTC)

Distribution of primes

This chapter is so shallow. I wish I knew more about it so that I could improve it but I don't yet. This chapter provides practically nothing, just a couple of quotes. Shouldn't here be something, like x/log x stuff? I am eager to learn this but at the moment I'm afraid I cannot improve the article even though I'd very much like to. --Ketorin 22:38, 9 November 2007 (UTC)

And the quote are not all correctly attributed. The page says that Paul Erdős said "God may not play dice with the universe, but something strange is going on with the prime numbers. [Referring to Albert Einstein's famous belief that "God does not play dice with the universe."] That widely believed but is not true. Carl Pomerance proposed that as something Erdos might have said as part of a talk at the MAA/AMS Joint meeting. This January, again at the same meeting he pointed out this quote was misattributed the very next day in te papers after he gave his previous talk. It is a great quote, so leave it in saying "Carl Pomerance proposed that Paul Erdős might say ..." Hopefully your other quotes are right. I just happened to be at both of Carl's talks mentioned here.Nagahyde (talk) 14:13, 7 June 2008 (UTC)

[11] agrees with you but it would be nice with a reference to a reliable source if we are going to claim that something widely believed is false. Pomerance is reliable for this purpose but is there a published reliable source saying that Pomerance said this? PrimeHunter (talk) 23:23, 7 June 2008 (UTC)

Notation

The article has the following phrase:

whenever a divides bc for ring elements b and c...

In order to make it more readable, I tried twice to modify it to:

whenever  $a\,$  divides  $b\cdot c$  for ring elements b and c...

There are important reasons in today's mathematics to write operations explictly, but first of all it is easier on eyes, one sees what's going on at a glance. Two unfortunate wikipedists have changed the text back. Please, don't let them do harm, even if a minor one. Let's have a more readable mathematical wikipedia. -- Wlod 21:04, 10 November 2007 (UTC)

Nobody in real life who works with commutative or non-commutative rings or algebras uses the notation you claim is used. Look at Jacobson's Structure of rings or any standard text book on Commutative Algebra (Zariski and Samuel, Atiyah and MacDonald) or even a book on Algebra / Algebraic Number Theory (take your pick, eg Lang). Please stop being pedantic. And do not make personal attacks. --Mathsci 21:31, 10 November 2007 (UTC)

--Cronholm144 18:52, 16 November 2007 (UTC)

Primality of 1

The article says:

Until the 19th century, most mathematicians considered the number 1 a prime,

But Euclid didn't, did he? Is it possible that Euclid and his coevals didn't consider 1 a prime number simply because they didn't consider 1 a number? Michael Hardy (talk) 23:19, 11 December 2007 (UTC)

numerals

In #Properties of primes, the word "numeral" is used to mean what I would call "decimal expansion" or "decimal representation". Is this a valid use of the word "numeral"? — Arthur Rubin | (talk) 02:08, 12 December 2007 (UTC)

A quick read did not show me such usage. Can you please tell us where in Prime_number#Properties_of_primes such usage occurs? The reason why decimal is not used might be because the said property also applies to non-decimal numeral systems. Karl (talk) 10:17, 12 December 2007 (UTC)

The first sentence:

• The base 10 numerals for all prime numbers except 2 and 5 end in 1, 3, 7, or 9. (Numerals ending in 0, 2, 4, 6 or 8 represent multiples of 2 and numerals ending in 5 represent multiples of 5, due to the use of the base 10 (2 × 5) system.)

My browser's search cannot find any other instances in the visible text, but that doesn't mean there aren't any. It seems occassionally to select a frame at random to search. — Arthur Rubin | (talk) 14:54, 12 December 2007 (UTC)

It doesn't read at all well. I would prefer: "When written in base 10, all prime numbers except 2 and 5 end in 1, 3, 7 or 9. (Numbers ending in 0, 2, 4, 6 or 8 represent multiples of 2 and numbers ending in 0 or 5 represent multiples of 5.)" Mathsci (talk) 15:11, 12 December 2007 (UTC)

That sounds good to me, but I would include a wikilink in base 10. PrimeHunter (talk) 15:40, 12 December 2007 (UTC)

In Popular Culture

I have removed the "In Popular Culture" section for the following reasons:

• None of the facts within the section had a cite or reference from a third-party source. To quote WP:V: "Any edit lacking a reliable source may be removed". The section has been there for some months without any of this stuff getting backed up.
• WP:TRIVIA and Wikipedia:"In popular culture" articles both discourage the creation and maintenance of sections of that type. I would have integrated the various factoids into the article, had they not all been about offhand remarks to prime numbers in science-fiction films. It is my assertion that none of the cultural references are culturally significant, although I'd be perfectly willing to back down if WP:RS were to be provided to convince me otherwise on that point.
• On this very talk page there have been suggestions of removing the section for over a year that have not been actioned, yet which nobody has objected to.

I do apologise for not taking this to the talk page initially, as I thought the removal would be uncontroversial.

Lankiveil (talk) 14:23, 15 December 2007 (UTC)

Hello again. Thanks for posting here. This section has been in the article for ages and, because it is trivia, is not subject to the same very exacting rules as the mathematics in the article. It might also be there for feel good reasons (an antidote to math phobia), since this is probably one of the most read introductory mathematics articles on the WP. I personally would be sorry to see it go, since it humanizes the article a little more. Prime number is probably intended for a very general audience, who are not necessarily experts in mathematics. Mathsci (talk) 15:19, 15 December 2007 (UTC)

Before I wrote the above message, you made a very ill-advised second attempt to blank this section and were reverted by an administrator. Please do not try this again. You could help improve the references yourself, if you have time on your hands. Mathsci (talk) 15:37, 15 December 2007 (UTC)

I don't think it qualifies as "blanking" given that I fully explained in the edit summary what I was doing. Lets all not use such loaded and emotive terms and work together to work out a solution =). Lankiveil (talk) 06:15, 16 December 2007 (UTC).

The problem from my point of view is that most of the pop culture stuff really is trivial (an obscure novel, an unremarkable episode of Stargate, etc). If the article is "too scary" for non math experts (and having a postgrad degree in linguistics, I still couldn't get much past the "history" section, although the lead was quite comprehensible), then perhaps parts of it need to be rewritten so they're easier to read, not by having a random grab-bag of "Primes in Science Fiction" section tucked away at the bottom. MichelleG (talk) 01:05, 16 December 2007 (UTC).

I agree that a "Primes in popular culture" section doesn't improve the article. What about the prose of the article makes it hard to read? Is it the history section itself, or does it come after that? -GTBacchus^(talk) 01:17, 16 December 2007 (UTC)

Coming in as a non-expert, some of the more esoteric formulae towards the lower half of the page are completely incomprehensible to me. I've no doubt that they're right, I just can't make heads nor tails :-). The lead section is pretty good and pitched at my level, although I think the bit about "ring theory" and below would probably be better off integrated later down in the more technical information. I would also move the "Applications" part up the page a lot, I would imagine that it's more useful for a lot of people to know what they're used for, rather than having a pseudocode computer programme to find them. MichelleG (talk) 01:33, 16 December 2007 (UTC).

The article is roughly structured like a general public lecture on mathematics: a significant portion near the beginning of the article is intended to be intelligible to a general audience; later on, there is a synthesis of all that is known about prime numbers, which by its nature - involving as it does the Riemann hypothesis and the Riemann zeta function- will probably not be intelligible to someone without special training in mathematics (the same applies to Encyclopedia Brittanica articles on mathematics); and somewhere in between there is a joke, to stop people falling asleep. You have removed the joke. Unless editors have been trained in mathematics, it would not seem clear that they can give a balanced overall view of the article. Many editors and even administrators on WP are unfamiliar with the wikiproject on mathematics. (This does not apply to Charles Matthews.) The project has articles covering the whole mathematical spectrum and numbers at least one Fields medallist among its contributors. In this sort of article, all sorts of people contribute, usually adding rather than subtracting content. In my case for example I helped formulate part of a sentence describing some of Goldbach's contributions. This required sourcing a letter in Latin to Euler. That took a day or two to do: it involved exchanges on this page with the editor who posted the first version of the material. Pressing an erase button takes far less time or intellectual effort. The applications of primes are of course treated in a separate article on Public Key Cryptography, wikilinked in the article, and invoke the "esoteric" theory of elliptic curves over finite fields. The section on prime numbers in nature seems speculative: there is regrettably no mention of the three-toed sloth or even homo sapiens, with his four sets of five digits. Mathsci (talk) 09:28, 16 December 2007 (UTC)

I would like to return to a discussion of content. Marcus du Sautoy, one of the most successful popularizers of mathematics and a past presenter of the Royal Institution Christmas Lectures, has several sections on his prime website devoted to primes and the creative arts:

• primes in films
• primes in theatre - in particular Arcadia by Sir Tom Stoppard (a play that I myself saw on June 23rd 1993, just after hearing the announcement by Andrew Wiles of his proof of Fermat's last theorem at the Isaac Newton Institute).
• primes in books - the book of Carl Sagan discusses the use of primes to communicate with aliens
• primes in Music - in particular use of prime cycles in the music of Olivier Messiaen; the numbers three and seven also feature in the musical numerology of J. S. Bach, in particular in the third part of his Clavier-Übung devoted to the organ.
• primes in the visual arts - M. C. Escher has used primes such as 67 in his drawings.

A guide to further non-technical popularizing books on primes and integers might also be useful (for example books in the style of "Gödel, Escher, Bach"). Mathsci (talk) 10:25, 16 December 2007 (UTC)

That's an interesting set of links, and maybe a link to [12] somewhere on the page would not go amiss. Lankiveil (talk) 10:08, 17 December 2007 (UTC).

The deleted section on films seems to have been taken from this website/book The Music of the Primes, which already appears as a link. The book is considered to be one of the best popularizing books on mathematics. In fact it won du Sautoy the prestigious Sartorius prize. [13] Perhaps it might have been more advisable to have been guided in these matters by du Sautoy's book, considering his exceptional experience in presenting this material to general audiences of all ages. Apart from his academic affiliations, he is a newspaper journalist and BBC television presenter (despite the almost insuperable visual problems created by his trademark tartan trousers).

Actually one of the main things that seems to be missing from the article at present as far as serious mathematical content is concerned is a brief and carefully written synthesis of the Role of prime numbers in analysis and geometry. This vista might start with the ideas of Andre Weil and John Tate on p-adic Fourier analysis, culminating in a description of the Langlands programme and the ideas of arithmetic geometry; this way of thinking figures in the proof of Fermat's last theorem. This could be probably be done succinctly in layman's language, with a set of wikilink pointers to other articles, using Daniel Bump's book on automorphic forms as a guide. However it would not be an easy task. Mathsci (talk) 15:36, 17 December 2007 (UTC)

I have inserted a new section into the article using that article as a basis - the new version is in proper prose rather than being a hodgepodge of dotpoints referencing soft sci-fi television programmes. I think it's definitely a topic worth covering, just not in the way that it was before. Mathsci, what is your opinion on the new section and can you think of any way in which it could be improved? Lankiveil (talk) 04:08, 22 December 2007 (UTC).

What you have included is very poorly written. The two examples you have selected from the list I provided are wholly inadequate and uninformative. You should try to find actual references to the piece of music by Messaien (which compositions for example?) and say how primes are used. You should include explicit references to the books of du Sautoy, Sagan, etc. Your sentence on Sagan is ambiguous: did you intend to say that he worked out his ideas for communicating with aliens with Frank Drake and then later included them in his book? If so, that is not what you wrote. The title of the section should be "Prime numbers in the arts and literature". At the moment it's just about one or more unidentified pieces of 20th century music and one popular science book on one or more unidentified aliens. Why not read the actual source books? There is for example:

• The Messiaen companion', ed. Peter Hill, Amadeus Press, 1994. ISBN 0-931340-95-0

From the index at the back of my copy of the book, it seems that Messiaen's use of prime numbers to create "ametrical music through natural phenomena" is discussed on pages 36-7, 39 and 356. In particular in Neumes rythmiques, one of the Quatre études de rythme (1949-50), Messaien used each of the primes 41, 43, 47 and 53 as a "nombre premier en rythme non rétrogradable". Prime numbers were thus used by Messaien to produce unequal rhythmic divisions that were neither repetitive nor straightforward (e.g. motifs of 5 or 7 rhythmic units, running concurrently). As he said himself "it is music inspired by the movements of nature, movements of free and unequal durations". Prime numbers figure also in one of hist most popular and aptly seasonal works, La Nativité du Seigneur (1935) for organ solo, especially in les anges and dieu parmi nous, a great toccata.

You are not correct in saying that what you deleted was « a hodgepodge of dotpoints referencing soft sci-fi television programmes ». It was I believe the list of films drawn from du Sautoy's book and website, not soft sci-fi TV programmes. Why did you write this? Improving the writing style would have been no problem. Your (sneering?) remarks could have been overlooked, if your proposed replacement had shown the slightest hint of scholarship. Mathsci (talk) 07:05, 22 December 2007 (UTC)

I have also just dug out my copy of Arcadia by Tom Stoppard

• Tom Stoppard, Arcadia, Faber and Faber, 1993. ISBN 0-571-16934-1.

Excerpts from the play were performed at Hertz Hall in the University of California, Berkeley in an event run by MSRI, Berkeley. The event included an interview between the playwright and Robert Osserman, deputy director. (The director at the time was David Eisenbud.) [14] A video of the event has been made available by MSRI. [15]

The play has various mathematical themes discussed in Chapter 10 of

• The Cambridge Companion to Tom Stoppard, ed. Katherine E. Kelly, Cambridge University Press, 2001. ISBN 0521645921

Here is an excerpt from page 2 (Act One. Scene One):

---

THOMASINA (aged 13): Is it a sin?

SEPTIMUS (her tutor): Not necessarily, my lady, but when carnal embrace is sinful it is a sin of the flesh, QED. We had caro in our Gallic wars – "The Britons live on milk and meat" – "lacte et carne vivunt". I am sorry that the seed fell on stony ground.

THOMASINA: That was the sin of Onan, wasn't it, Septimus?

SEPTIMUS: Yes. He was giving his brother's wife a Latin lesson and she was hardly the wiser after it than before. I thought you were finding a proof of Fermat's last theorem.

THOMASINA: It's very difficult, Septimus. You will have to show me how.

SEPTIMUS: If I knew how, there would be no need to ask you.

....

THOMASINA: If you do not teach me the true nature of things, who will?

SEPTIMUS: Ah. Yes, I am ashamed. Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat's last theorem, by contrast, asserts that when x, y and z are whole numbers each raised to the power of n, the sum of the first two can never equal the third when n is greater than 2.

(Pause.)

THOMASINA: Eurghhh!

SEPTIMUS: Nevertheless, that is the theorem.

THOMASINA: It is disgusting and incomprehensible. Now when I am grown to practice it myself I shall never do so without thinking of you.

---

BTW, as Thomasina immediately grasped, the theorem has only to be checked when n is prime :) Mathsci (talk) 08:36, 22 December 2007 (UTC)

Well, what is there is only a starting point, and it's ripe for expansion, of course. Lankiveil (talk) 00:58, 23 December 2007 (UTC).

Make pseudocode more accessible to non-programmers

I would like to see the pseudocode for finding prime numbers to be reworked and more understandable to people who are not rock-star programmers and mathematicians.

Currently it is:

// arbitrary search limit
limit ← 1000000

// assume all numbers are prime at first
is_prime(i) ← true, i ∈ [2, limit]

for n in [2, √limit]:
    if is_prime(n):
        // eliminate multiples of each prime,
        // starting with its square
        // 2n, 3n, …, (n-1)n already eliminated
        // nn, (n+1)n, (n+2)n, … to be eliminated
        is_prime(i) ← false, i ∈ {n², n²+n, n²+2n, …, limit}

for n in [2, limit]:
    if is_prime(n): print n

I would like to see it changed to something more like this, where each instruction is simply explained in English. I've only changed a few lines (the ones I understand), and put variable names in parentheses. The lines I haven't changed, I don't fully understand. Can anyone help rewrite this?

// arbitrary search limit
set an arbitrary limit (limit)

// assume all numbers are prime at first
// instead of saying "assume all numbers are prime...", maybe it should say 
// create an array of all numbers from 2 to the limit (limit), and flag each one
// as a prime number, beginning with the assumption that all are primes.
is_prime(i) ← true, i ∈ [2, limit]   // What does the i stand for?

for each number (number) between 2 and the square root of the limit (limit), inclusive
    if is_prime(n):
        // eliminate multiples of each prime,
        // starting with its square
        // 2n, 3n, …, (n-1)n already eliminated
        // nn, (n+1)n, (n+2)n, … to be eliminated
        is_prime(i) ← false, i ∈ {n², n²+n, n²+2n, …, limit}
end loop

for each number (number) between and the limit (limit)
    if (number) is in the array (is_prime), then print it ??
end loop

Thanks. LinguistAtLarge (talk) 00:52, 16 January 2008 (UTC)

Suggestion: Remove pseudocode

I saw the above section and would actually prefer to remove the pseudocode completely from this article (and I have written dozens of prime sieves so it's not due to lack of understanding or interest). It was discussed in May 2006 with no consensus at Talk:Prime number/Archive 4#Program. I didn't participate then and consensus can change so I suggest taking it up again. My arguments: 1) This is the main article in Category:Prime numbers which has more than 100 articles including subcategories. There shouldn't be so much detail about sieves which are a small part of prime numbers in this already long article, and pseudocode may scare some readers away. 2) There is already pseudocode for the same algorithms in Sieve of Eratosthenes and Sieve of Atkin. Improvements to pseudocode should rather be there instead of making another pseudocode here. 3) Pseudocode fits the subjects of the sieve articles perfectly, there is more room in those short articles, and the algorithms are already discussed more there which should give easier understanding of the related pseudocode.

If there are still editors who want different pseudocode for the same algorithms in two articles then I suggest a compromise: Move it from Prime number to Generating primes where it fits the subject more closely and there is much better room. So what do people think: Keep, delete, or move the pseudocode? PrimeHunter (talk) 02:12, 16 January 2008 (UTC)

I agree that the pseudocode belongs in the individual sieving articles, not here. —David Eppstein (talk) 02:25, 16 January 2008 (UTC)

• Move and Make readable please. LinguistAtLarge (talk) 04:37, 16 January 2008 (UTC)

I have removed the pseudocode in [16]. PrimeHunter (talk) 01:14, 3 February 2008 (UTC)

Sun's conjecture

I have removed the following:

• Sun's conjecture: Each natural number $n\not =216$ can be written in the form $p+T_{m}$, where $p$ is a prime or zero, and $T_{m}=m(m+1)/2$ is a triangular number. (See: http://arxiv.org/abs/0803.3737 )

My edit summary [17] was:

revert edits by 221.224.13.179. Section says "famous conjectures". This conjecture was aparently selfpublished a month ago with no sign of fame

I actually knew the conjecture but only because I subscribe to an email list where Sun posted it.[18] PrimeHunter (talk) 23:48, 2 May 2008 (UTC)

The sum of prime numbers < n

I have reverted the addition of the section "The sum of prime numbers < n ~ n^2/(2log(n)-1)" [19] by Hillcino368. My edit summary [20] was "revert edits by Hillcino368. Looks like WP:COI and WP:OR from mails by hillcino368 at http://tech.groups.yahoo.com/group/primeform/message/8977. Was also poorly formatted and too detailed". PrimeHunter (talk) 01:46, 7 June 2008 (UTC)