Talk:Prime number

Links: http://www.utm.edu/research/primes/

The first 750,000 primes: http://www.geocities.com:80/ResearchTriangle/Thinktank/2434/prime/primenumbers.html

Now here's an interesting question which may even provide a valid use for subpages.

Would adding a list of all the prime numbers known to mankind be counter-productive to the idea of the Wikipedia? It is "a compendium of human knowledge", regardless of how obscure and arcane.

(I suppose one could extend this to listing Pi to a quadrillion digits, to be somewhat hyperbolic.)

Thoughts? --Colin dellow

I'd say that the encyclopedia should be a compendium of all useful knowledge. Digit number 323454 of Pi and the temperature at noon on May 7, 1976 in Salt Lake City are part of human knowledge, but really, nobody has a use for this information except maybe some highly specialized experts, who know where to look it up.

I perhaps didn't make my point clear. Wikipedia presents the ability to have "useless" trivia because of the Wikipedia is not paper argument.

Perhaps it will be useful only to a specialised group, but does it hurt to have it? It would increase the amount of information available in the Wikipedia. (I found the comment about temperatures in Salt Lake City at a given year to be somewhat unrelated to this question, BTW.) --Colin dellow

Actually, I don't think it would be possible to list all the known primes. There are just too darn many of them, and it's too easy to find more. - Hank Ramsey

Note regarding the proof of the infinitude of primes: we use the fact that every composite number has a prime factor. I think that requires proof in itself.

Yes, perhaps, if it isn't obvious. It is clear that if a given number has a composite divisor, then the divisors of that divisor are also divisors of the original number (if A=B*C and C=D*E then A=B*D*E). If any of these divisors are composite, we can apply this principle recursively to find more divisors of the original number. We cannot recurse indefinitely because the number cannot have an infinite number of divisors. So eventually we must reach a point where all the new divisors are prime. QED

sub page or article on infamous research project "A Short List of Even Primes" which is of course, mainly acknowledgements and academic scaffolding, followed by the list, viz. 2

I wouldn't mind a link to a joke if it's clearly labeled as such. For that matter, I wouldn't mind adding the old "proof" that all primes are odd. -LC

The article says:

There are infinitely many prime numbers. The oldest known proof for this statement dates back to the Greek mathematician Euclid. It is also one of the oldest known proofs by contradiction.

I thought that Euclid stated the theorem as "Given any prime, there is a larger prime." With the theorem stated like this, the proof isn't a proof by contradiction -- the existence of the larger prime is shown directly. Can someone confirm this, or am I misremembering? --Zundark, 2001 Oct 11

I think you're right; I'll take the contradiction bit out. --AxelBoldt

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!/ ∏(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 Križanič 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.

---

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)

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 √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

--

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 ∏(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 ∏(1234567890) by hand. I don't believe that even Gauss would solve in one hour ∏(100) as he did Σ(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!>∏(n). (Is this proven?) Are there any? I've found trivial 3!± 1 = ∏(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 p_k+1, rather than the initiating prime p_k. >>

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 p_n-p_n-1=g_n+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 p_{X(of the 64th gap)}-p_{Y(of the 63th gap)}-1=p_Z=1693182318746371-1686994940955803-1=6187377790567. It is correct p_X(n)-p_X-1-1=g_n-1 as it is in this first three examples: p_{2(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 pensil 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 g_n refers to the fact that g_n 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, g_x1=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)

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.

Talk:Prime number

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