Talk:Prime number/Archive 2

Types of primes
I took this out for now:
 * multifactorial, compositorial or binomial primes

since the terms are not defined. I couldn't understand the examples
 * Multifactorial prime is for example 43328! · 7 - 1, compositorial prime 8711!/ &prod;(8711) - 1 and binomial prime 104087!/(52043! ± 52044!) + 1.

And in this paragraph
 * and can have many other properties in collecting of different terms or can be derived from some well known sequences as for example as the primitive part U(n) of the nth term in the Fibonacci sequence e.g. U(40295) or as V(n) in the Lucas sequence e.g. V(25504) and many more.

what is the "primitive part" of the nth term in the Fibonacci sequence? AxelBoldt

Yes, Axel I'll soon make those definitions more clear. And on the other hand if you understand the first two definitions (primorial and factorial prime) you should understand the next three ones, because they are just futher extendings of the first two ones. These short contributions stole me some 3 to 4 hours of intense work just to get them together and one can easily put them out in a minute. The topic of pure and just pure primes is for me at a highest interestings. I think Axel you're looking for too much "usefull" and wonderful definitions, theorems, proofs and such inhere. Math is not just that. The good example for this is for instance (a pure mathematician) Keith Devlin with his piercing work for better understanding of the whole past, present and future math or I can say this for our mathematician France Kri&#382;ani&#269; who wrote some nice Devlinlike books - and he is still writting them. For example integral in a complex is not for a high school, but he had put it in his huge textbook for secondary schools. He put there some beautifull work of theoretical astronomer Möbius as well. I can talk on and on - but I am afraid someone would say in Marley's manner I've got so much things to say, ha, ha. XJam keep on moving and try putting some more stones in math knowledge on this icy road. For the term "primitive part" I need more Time because nobody told me about, hey ho. (Back to work XJ again, - come together and make it work, whoahh, we got five days to go, working for the next day, hey hey, now ... [Bob Marley, Work, live at final tour in Berlin 1980 ]). And I have enough Time because I really do not want to steal anything from Nature. I do believe that Fibonacci never dreamed that someday someone would talk some more about his "obscure" sequence found everywhere in a Nature itself. That goes for Lucas (and many, many others) too. I hope I'll achieve at least fair level of Axel's rigorousness soon.

Another thing (I'll say this as fast as I can :-) ) as of the first above external link someone put in this talk. I'll generate with that list an Ulam-(Möbius) cloth and I'll post a picture of it in Wikipedia's digital archives as soon as possible. We can then Distiquence Arithmeticaenniolus (this is a weird Gausslike verb) some more, if ya agree. This list is for me very Hardylike "usefull" for me, because I have no current working algorithm to produce it. But if someone had already made an Ulam cloth of primes, please let me (us) know. XJam [2002.03.23] 6 Saturday (0) - the 3rd day of spring 2002. Natty Dread 20000 miles away from home.

Largest known primes
This sentence appears under the heading 'Largest known prime': "This result with purely PC based computer with < 1GHz Pentium processor beat some previous prime-runners as supercomputer Cray T94 was." I can't be certain as to the meaning of this sentence, making it rather difficult to fix. Anyone?


 * Yes Ostrich as it is evident from your user profile page you're already in computer's world so you should understand that (almost simple) computer's net fight of David against giant Goliath. I would not refuse one such model T94 if it would by a New Year's Day's present but anyway. You should check at fist GIMPS term to see how a small or a huge connected computers via the net can work very efficiently together. We can say that Wikipedia works in the same way. There are many examples (I know not all of them) of such work: Odlyzko's mathematical work with te Riele on Mertens conjecture (see for the current wikipedia status of this at http://www.wikipedia.com/wiki/Talk:Moebius_function, SETI@home, ISDN-ODETTE conection of European automobile industry and many more) It is good to know Odlyzko and te Riele used for the job one of the "first" models of Crays - a model 1 (It sounds just like a fomous Ford's model T1). And probybly Odlyzko had decided to make some of his outstanding calculations on such dispersed "systems". He wrote some article on this subject too. As you like fractals can you give me any help on my previous instance about Ulam's cloth of primes? And actually it was exactly 500 MHz Pentium based PC to gain that recorld on the biggest (Mersenne) prime up to date. I hope this will kill the curiosity in cat. Greets to Ox from Ce. --XJam [2002.03.23] 6 Saturday (1st ed)


 * Sorry folks - above - I was talking nonsenses about Ulam's cloth. Ulam's prime spiral is as it is. I have already put its representation on wikipeda (see mobius function) so I don't have to make it again but anyway I do believe that mentioned list of primes is very important for everyone who wants to deal with primes. --XJam [2002.03.24] 0 Sunday (0)

Someone should update this section. . . . The 42 Mersenne prime has now been found (February 2005).

Gaps between primes
I moved the following material here for now:


 * For example the gap bettween 1693182318746351 and 1693182318746371 is g = 1693182318746351 - 1693182318746371 - 1 = 19, but the next consecutive gap g64 is 1131. This kind of gap is called maximal prime gap with property of occurences of gaps of at least of this length following initiating prime p and is the largest known maximal prime gap, found by T. Nicely and B. Nyman in 1999.


 * Jose Luis Gomez Pardo in 2001 found the largest first known occurence prime gap greater than 1131, which is 21611, following the initiating and certified prime 10999 + 24196831 but with its consecutive prime to be probabilistic, that is, proven to be statistically prime (with a probability extremely close to 1), using, for example, Miller-Rabin test with multiple bases or some other primality test. There are of course many prime gaps with sizes from 1131 to 21612 but 1131 is the maximal one. This will probably change soon. Cramér showed that if the Riemann hypothesis holds, it would be:

g < k &radic;p log p.

Specifically, I have the following questions: what exactly is a maximal prime gap. I do not understand the example and explanation given above.

Second, it seems that Pardo did not infact find the largest gap, but just a probable gap, is that correct?

Third, if there are many gaps with sizes between 1131 and 21612, how can 1131 be the maximal one? AxelBoldt

Primorials
The largest known Primoral seems to be outdated; mentions 392113#+1

Let me say these things:

//-1 Yes, strictly speaking primorial primes are not special case of factorial ones. Generating a 'prime product' for n as &prod;(n) goes in 'a same manner' as a function n! = 1 · 2 · 3 · 4 · 5 · 6 · ..., but we have to 'check' first its argumets for primality what is a bit harder than adding a number by 1. Thus 1 · 2 · 3 · 5 · 7 · 11 · ... is easy just for first numbers of n. Try to get &prod;(1234567890) by hand. I don't believe that even Gauss would solve in one hour &prod;(100) as he did &Sigma;(100). Another strictly mathematical question would be which primorial primes are members of factorial prime set or vice verse as for arbitrary n is n!>&prod;(n). (Is this proven?) Are there any? I've found trivial 3!± 1 = &prod;(3) ± 1 = {7,5}, but here Nash's logic already ends...


 * Primorial primes can never be factorial except in the two cases you list. The reason is that all factorials beyond 3! are divisible by four, but products of different primes are never divisible by four. AxelBoldt

//0 You didn't move just above material but you moved out also this: << This gap was discovered by Euclid in 300 B.C.. Others define it to be simply G = q - p, so the gap G1 following the prime 2 has the length 1. Another definition for gap is with a parameter r = g/2 and some other authors have specified a gap by the terminating prime pk+1, rather than the initiating prime pk. >>


 * Yes, the slight variations in the definition of prime gap didn't seem to be important; the concept itself is more important. What exactly did Euclid discover? AxelBoldt

Euclid was probably first one who was thinking about gaps, so that's why I had put him in the article. I've noticed that someone sometimes wants definitions and nothing more than definitions and sometimes not. I think I have explained also good enough what initiating prime and its 'bounded' partner are. Gaps are very close connected with primes, so they should be well defined for better understanding of primes themselves.
 * Have no futher knowledges about Euclid's discovery.

//1 The example explains the ambiguity of maximal gaps. There are no gaps with greater size than 1131 from 2 to initiating prime 1693182318746371. So 1131 is the largest one bellow this prime and it stands on 64th place. g1=0 is first one. I don't know if a list of all known maximal gaps is appropriate for the article? We can put it in, if someone wants.


 * I see, that makes it clear. I'll put the maximal gaps back in. One more question: you say above that ''g64 = 1131. Does that mean in your notation that the 64th prime minus the 63rd prime + 1 equals 1131? AxelBoldt


 * No, no. Indexes of gaps and primes here are not equal at all. Perhaps 'late enough' would be pn-pn-1=gn+1-1 for some period or (it would be infinitive) after some initiating prime p. In this case is: g64-g63+1=1131-923+1=208. But it is here pX(of the 64th gap)-pY(of the 63th gap)-1=pZ=1693182318746371-1686994940955803-1=6187377790567. It is correct pX(n)-pX-1-1=gn-1 as it is in this first three examples: p2(of the 2nd gap)-p1-1=g1=3-2-1=0,p4(3)-p3-1=g2=7-5-1=1, p9(4)-p8-1=g3=23-19-1=3 and so on). For now on we need pencil and paper to produce more examples. And after some time we would need a computer soon. And finally 923 is 63th maximal gap. Initiating prime for g64 is 1693182318746371 but is not 64th prime.


 * Oh, so the n in gn refers to the fact that gn is the n-th maximal gap? Then we have to change the notation in the article. AxelBoldt

//2 (Probably) Yes.

//3 I think gaps above 1131 were found in this way. Someone took some special types of primes and he calculated with one primality test gaps between them. Some were greater than 1131 but these primes were much bigger and much 'far away' than primes near 1 &middot 1016. Someone should examine all primes above initiating prime of the maximal gap g64. I do believe that there exist a gap, let us say, gx1=24242824248748732872000000000000000000000001, but, first nobody knows where and second, if he knows it, what would this help him to complete a gapopedia. And finally what are the still uknown properties of gaps we should look for? (I do hope I had understood Pardo's discovery and of others well enough. --XJam [2002.03.27] 3 Wednesday (0)

Defining 'prime number'
The first section gives two contradictory definitions of prime. The first definition would imply -2 is not prime. The article should either give a single definition, or it should explicitly say that there are two different definitions in common use. The math dictionary I own gives only the second definition: an integer not equal to 0, 1, or -1, whose positive divisors include only 1 and itself.


 * The first section gives two definitions of prime, but they aren't really contradictory. One is a definition of primes in N, the other of primes in Z. Perhaps this should be made clearer. If the term "prime numbers" is used without further clarification it usually means primes in N, and I think this is how the term is usually used in Wikipedia. --Zundark, Thursday, April 4, 2002


 * Yes the first definition is a bit deficient. In the second section is then said that primes can be negative natural numbers too. (In fact 1 is prime, but this is a convention, right?) How about 0? Why 0 is not a prime? We can't make natural factorization on it perhaps this would be too trivial. --XJam [2002.04.04] 4 Thur's day (0)


 * By the fundamental theorem of arithmetic, -2 cannot be prime, since (-2)&sup2; = 2&sup2; = 4. You also can't prime factorise -1 (and you can't uniquely prime factorise any negative number, or 0). Unless someone can demonstrate the usefulness of 'negative primes', implying 'prime natural number' every time you say 'prime' seems superfluous. We can also use the theorem to define primes P as the increasing sequence of integers {p1,p2,p3,...} such that for all n &isin; N there is a unique sequence of integers X {x1,x2,x3,...} where p1x1</SUB></SUP>·p<SUB>2</SUB><SUP>x<SUB>2</SUB></SUP>·p<SUB>3</SUB><SUP>x<SUB>3</SUB></SUP>·...=n (i.e. the primes are exactly those integers which can uniquely prime factorise every natural number). We can then show 1 isn't prime (or the prime factorisation of 1 wouldn't be unique, and therefore the corresponding set X has all elements set to zero), and 2 is prime, and no prime can be negative, etc (you can probably also show that there is only one P which satisfies these conditions, but I haven't proved that in my head yet). (I allow for xi to be negative, but that still results in only one set of primes, which is trivial to prove. We can also use n &isin; Q, since rational numbers must also have a unique prime factorisation with some negative xi) Elektron 11:34, 30 Apr 2004 (UTC)


 * I've also corrected it and made it prettier and added bits. Elektron 12:14, 30 Apr 2004 (UTC)


 * On the other hand, for a unique factorization domain and its appropriate definition, &minus;2 is a prime; because then uniqueness is defined up to multiplication by a unit in the ring. Charles Matthews 12:08, 30 Apr 2004 (UTC)


 * That went completely over my head Elektron 12:14, 30 Apr 2004 (UTC)


 * The point is that we have a concept of prime elements in any integral domain. The integers form an integral domain whose prime elements are the negative primes and the positive primes. The Fundamental Theorem of Arithmetic can be restated for this more general concept of primes, and it applies to all unique factorization domains. --Zundark 12:44, 30 Apr 2004 (UTC)


 * Right - but prime number really ought to stick to natural numbers, anyway. Other stuff should be done by links. By the way, this page pretty well takes the biscuit for being a confusing talk page. Anyone here competent and able to clean it up/archive stuff, without infringing any wikiquette? Charles Matthews 15:04, 30 Apr 2004 (UTC)


 * You can uniquely prime-factorise 1 (2<SUP>0</SUP>3<SUP>0</SUP>5<SUP>0</SUP>...), but you can't uniquely prime-factorise 0 or -1 no matter what you define as 'prime'. You can uniquely prime-factorise integers less than -1 (using the primes -2,-3,-4,-5,-6,-7,-9,-11,-13), but this limits you to an odd number of prime factors for every integer. My point is this: each natural number is uniquely represented by a sequence of prime indices (e.g. 1 ~ {0,0,0...}, 10 ~ {1,0,1,0,0,...}, 12 ~ {2,1,0,...}), and each sequence of prime indices uniquely represents a natural number. Extending this to zero or negative numbers doesn't work, and changing the fundamental theorem of arithmetic so that no 'uniqueness' is necessary seems icky. Elektron 09:19, 1 May 2004 (UTC)


 * OK, perhaps it isn't so hard to understand, for example, that unique factorisation of polynomials is naturally stated as 'up to constants'. In any case anyone who wants to know the standard usage on unique factorisation in more abstract contexts has to get over this hurdle. Charles Matthews 10:39, 1 May 2004 (UTC)


 * Elektron, I just wanted to re-assure you that what Charles and Zundark are saying is standard undergraduate mathematics (and has been for decades) and not some wacky left-field thing. "Uniqueness up to" is a pervasive and undoubtedly useful concept in algebra, although it may seem a little odd when meeting it for the first time. Pete/Pcb21 (talk) 09:06, 4 May 2004 (UTC)

-1, 0, and 1 are not prime, by definition, because that definition is more useful than one that includes them. --LDC


 * When discussing primes it has always been the convention to assume we are only discussing primes in N: this is not a Wikipedia convention, it is found in all number theory AFAIK. When discussing negative primes (those not in N but found in Z) people will usually clearly identify this as such. Most mathematicians do not regard negative primes as particularly interesting, however one day someone may discover some fascinating property exclusive to negative primes and change this attitude.


 * The definition of 1, 0 and -1 as neither prime nor composite occurred because (as LDC rightly pointed out) there was no perceived benefit to a prime definition that included them, and it messed up an otherwise very useful definition. Until someone can prove that there is a genuine benefit to changing this definition (which has been around for several hundred years) this will probably remain the case. The quotient of 1/0 is left as "undefined" for similar reasons - having a defined value for 1/0 only messes things up and doesn't serve any (identified) useful purpose.


 * A request - could someone (Axel, LDC, other?) write the article on illegal primes? It sounds really interesting and I was keen to learn the rest of the story. - User:MMGB

The reason that 1, 0, -1 are not considered prime nor composite can be best explained by thinking in terms of the concepts of general ring theory. We are looking at a commutative ring with unity. Prime elements in these rings are usually defined as elements p such that whenever p divides ab, then either p divides a or p divides b. EXCEPT that we exclude those elements that trivially divide everything in ring, in the case of the integers, these elements are 1, -1, which are called the UNITS of the ring. A prime is defined as a non-zero, non-unit, that satisfies the above property. 1 and -1 are excluded because they trivially divide everything. Zero is excluded because it only divides itself, nothing else. The reason the NEGATIVE primes are considered not so interesting is that we only really care about primes "considered up to a unit", i.e. primes which differ by a unit element are identified or associated with each other. Since 1 and -1 are the only units, this means every class under this relation has 2 elements, namely itself and its opposite, -3 isn't important because it's ALREADY identified with 3 by multiplication up to the unit element -1.


 * The Shorter OED definition includes 1 as a prime number. Also, since every definition of "prime" explicitly excludes 1, one is led to believe that it really ought to be anyway. Peter T.S. 02:33, 5 November 2005 (UTC)


 * I suggest the following shortest possible and correct definition of prime for the Wikipedia article "Prime number":
 * A prime number is a natural number with exactly two natural divisors.
 * Hans Rosenthal (hans.rosenthal AT t-online.de -- replace AT by @ )


 * PS: Never forget that here we are just seeking for a fine and simple definition of the term prime number !

The definition given in the first paragraph is contradicted by the information given later on primes in rings, which is more modern. A link in the first paragraph to the more general definition would be helpful.

Re: "When discussing primes it has always been the convention to assume we are only discussing primes in N" This clearly does not correspond to the page as it stands. Nor should it, entirely. I am coming here from a discussion in which the definition from this page was cited inappropriately, based on its first paragraph alone. So what is up currently is misleading. The question "is -2 prime?" is fairly subtle; the answer is: if you are in fact working with all integers, then in that context -2 is indeed prime; and if you are working only with positive integers, then the question does not arise. Abu Amaal 19:58, 26 February 2006 (UTC) [moved to proper subsection, condensed] Abu Amaal 15:13, 27 February 2006 (UTC)

0 is not prime. It is divisible by every integer.

In mathematics one should strive to have as little as possible definitions. If something follows from the definition it should n&oacute;t be &iacute;n the definition. The definition given by Hans Rosenthal, A prime number is a natural number with exactly two natural divisors, is the shortest and most accurate definition possible. That 1 and the prime number itself are its only (Natural) divisors logically follows from this definition. After all:
 * $$1 = 1 \times 1$$,

therefore:
 * $$\frac{1}{1} = 1$$.

Since we know that $$\forall_{n\in\mathbb{N}}\left [ n \times 1 = n \right ]$$, it goes without saying that:
 * $$\forall_{n\in\mathbb{N}}\left [ n \times \frac{1}{1} = n \right ]$$.

And thus it follows that for every natural number, 1 is a divisor. Something similar can be done for the prime number itself. Because it follows from the definition that, if a natural number has exactly two (different) divisors, these divisors are 1 and the prime number it self, this should n&oacute;t be included in the definition.
 * Ruud

Ulam spiral
According the the article, the Ulam spiral question is an open one. However, there is information about the origin of this pattern in M. Gardner, Sixth Book of Mathematical Games, Scribner?s, 1971 if someone would look it up. Anyhow, one quick explanation is that any prime (excluding 2 and 3) is equal to a multiple of 6 +/- 1. The reason for this is as follows: Break the primes into 6 columns (http://www.geocities.com/~harveyh/Image_No/Plot_Lin.gif, from http://www.geocities.com/~harveyh/moreprimes.htm . Anything in column 2, 4, or 6 is divisible by 3.  Anything that is in column 3 is divisible by 3.  Those numbers form the basis for diagonal patterns when using a spiral since the spiral offsets these numbers by 1 in each loop (looking at one particular side of the loop).


 * Have also a look at the discussion in Talk:Ulam_spiral. There is probably not much mystery left in Ulam's spiral. FvdP 11:53 Nov 19, 2002 (UTC)

Formulae for primes
Here are the explicit formula for prime numbers, twinprimes, number of primes, number of twinprimes. It was published in the Proceedings of the Indian Academy of Sciences and reviewed by other mathematical journals. The formula was examined as correct by world renowned mathematicians such as Dr.Paul Erdos, Dr.Halberstien and Dr. K.Ramachandra of TIFR. The four formulae were discovered by Venu Atiyolil in the year 1983 and are presented below in the pdf format. The proof may be difficult to follow to some mathematics students as the method of writing was from a world class School of Mathematics where the author was a research member. http://www.ias.ac.in/jarch/mathsci/92/00000050.pdf http://www.ias.ac.in/jarch/mathsci/92/00000051.pdf http://www.ias.ac.in/jarch/mathsci/92/00000052.pdf http://www.ias.ac.in/jarch/mathsci/92/00000053.pdf http://www.ias.ac.in/jarch/mathsci/93/00000068.pdf

<End of article> ____________________________________________________________________

I was going to change the formulas to TeX, but some of them don't seem to make sense.

Under "Formulas generating prime numbers", the first "alternately" definition given subtracts (j-1)! from itself. Thus the summation function is just 1/j.

The second "alternately" divides j by itself. This also effectively means (j-2)! is subtracted from itself, so the summation function is 0.

These are obviously incorrect. What exactly is to be done about these? Eric119 06:30 Feb 17, 2003 (UTC)
 * Um, never mind this. Silly me, I didn't catach on that [x] was given as the floor function. I thought it was just normal bracketing. Okay there are no problems with the formulas now (except I still need to convert to TeX...). Eric119 06:41 Feb 17, 2003 (UTC)

-


 * why does composite link to Prime? --PY

An anon user added some bogus about prime number formulae. Please do note that the number of computations in these formulae is at least as big as the number of computations involved in optimised versions of Erastothenes' sieve for a number of comparible size. Can someone make it NPOV? Gebruiker:Dedalus 12:24, 28 Feb 2005 (UTC)


 * I've reverted the page, and I'm moving the information about Venu Atiyolil's contributions to formula for primes, where I will rewrite it NPOV. -- Dominus 14:47, 28 Feb 2005 (UTC)


 * I replaced the link to the author's GeoCities site (which was unreliable, since the paper in question was only downloadable at bitmap images that soon exhausted his bandwidth quota) with links to the PDF archive version of the pages on the official web site of the Indian Academy of Sciences. -- Dominus 15:09, 28 Feb 2005 (UTC)

I've reverted all the changes by this anon guy. He keeps adding info to articles when it's been moved from one article to another, and he also has been deleting/altering comments on talk pages. CryptoDerk 03:13, Mar 5, 2005 (UTC)

The two classic formula $$n**2-n+41$$ and $$n**2-79*n+1601$$ return a better prime density than the asymptote $$n/log(n)$$ thru the 1st 9999 integers unaltered. One would expect approx. 2500 primes (10000/log(10000))and each eqn yields approx. 4150.--Billymac00 21:55, 9 June 2006 (UTC)