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Archive 1 | Archive 2 | Archive 3 | Archive 4 | Archive 5 |
I took this out for now:
since the terms are not defined. I couldn't understand the examples
And in this paragraph
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?
Someone should update this section . . . . The 42 Mersenne prime has now been found (February 2005).
I moved the following material here for now:
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
The largest known Primoral seems to be outdated; [1] 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 ∏(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...
//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. >>
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.
//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.
//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 · 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)
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.
-1, 0, and 1 are not prime, by definition, because that definition is more useful than one that includes them. --LDC
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 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 nót be ín 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:
therefore:
Since we know that , it goes without saying that:
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 nót be included in the definition.
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).
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)
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 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 and return a better prime density than the asymptote 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)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 | Archive 5 |
I took this out for now:
since the terms are not defined. I couldn't understand the examples
And in this paragraph
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?
Someone should update this section . . . . The 42 Mersenne prime has now been found (February 2005).
I moved the following material here for now:
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
The largest known Primoral seems to be outdated; [1] 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 ∏(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...
//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. >>
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.
//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.
//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 · 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)
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.
-1, 0, and 1 are not prime, by definition, because that definition is more useful than one that includes them. --LDC
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 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 nót be ín 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:
therefore:
Since we know that , it goes without saying that:
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 nót be included in the definition.
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).
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)
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 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 and return a better prime density than the asymptote 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)