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Some page format added. Charles Matthews 07:49, 4 May 2004 (UTC)
It's me the only one (screen resolution: 1024 x 768 in a 15'' LCD) that, in this image added on August the 6th [1], can see nothing more than a reddish black rectangle? Maybe the color choice is great for a background, but no to illustrate this article. Maybe in black and white it could be used here, otherwise, I think it would be better to remove it. -- 79.150.126.120 ( talk) 13:31, 26 August 2008 (UTC)
I second that. Apart from the poor colour choice, it is patently useless to display a small thumb of a huge image in which every pixel is supposed to carry one bit of information. It results in a random meaningless blur. — Emil J. (formerly EJ) 14:01, 26 August 2008 (UTC)
All I see is a black rectangle. The other problem (but easily fixed) is that it's uncouth to use an asterisk for ordinary multiplication, as in 3 + (1281 * 1024 * 2). One should write 3 + (1281 × 1024 × 2). Michael Hardy ( talk) 18:14, 12 August 2009 (UTC)
There's little more besides a headline stating more information to come soon, but http://mersenne.org/prime.htm claims to possibly have found the 45th mersenne prime number. Slashdot has also covered it. -- 76.85.144.126 ( talk) 00:28, 28 August 2008 (UTC)
FYI: Now there is news of a 46th known Mersenne prime see Fox News reported Sept 27, 2008 —Preceding unsigned comment added by 203.171.45.66 ( talk) 21:31, 27 September 2008 (UTC)
I've removed some material beginning Chris Curtis, a computer scientist from Auckland New Zealand showed the first predictive method for prime numbers by showing the first complete pattern in a method also used on the same day to rationalise Pi. as possible WP:OR, WP:PEACOCK terms, unsourced and contentious. Richard Pinch ( talk) 11:27, 20 September 2008 (UTC)
Should Wilson's theorem be listed among the primality tests? I know it is inefficient, but it does show that a number can be tested for primality with only one division. Asmeurer ( talk ♬ contribs) 06:09, 20 November 2008 (UTC)
I have always been wondering whether prime numbers exist in nature some way or another. Take a look in this article at what Stephen Wolfram has got to say about the rings of Saturn (it is somewhere in the middle): http://www.kurzweilai.net/meme/frame.html?main=memelist.html?m=17%23646. Does anybody know more about that and can give some examples in the article? —Preceding unsigned comment added by 89.236.28.157 ( talk) 10:37, 31 December 2008 (UTC)
I put here a number of items that I may put to the see also section or the main text. I will undertake some revision and expansion of the text shortly. Jakob.scholbach ( talk) 19:59, 31 March 2009 (UTC)
The distinction between Prime and Irreducible is never made in this article. One reason for this is because the article addresses primality only in the natural numbers. Both of these issues should be addressed. For example in other rings such as , prime and irreducible are not equivalent as in . This is a very important distinction that is (as far as I can find) never made on Wikipedia.
If generalizing this article would make it too cluttered or lengthy, other articles about Prime and Irreducible should be created. —Preceding unsigned comment added by Silverhammermba ( talk • contribs) 15:40, 16 April 2009 (UTC)
Dear Sir:
I'm sorry, but the way that the phrase below, took from the text, was whritten, lead us to an ideia that Euclid cames after Mersenne.
"Euclid also showed how to construct a perfect number from a Mersenne prime".
Forgive me for any inconvenience.
189.106.2.214 ( talk) 02:26, 23 April 2009 (UTC)
I am curious as to why there are so many mentions of means to determine primality of numbers, when the classic means to find primes is not mentioned anywhere in the Prime Number article: the Sieve of Eratosthenes. The sieve is not even listed in the See Also section. William R. Buckley ( talk) 16:55, 24 May 2009 (UTC)
We have to painfully state that it is more important to know that prime lie on a line and predict that line which breakthrough may have already happened using inverse 19 , because there are infinite primes, we cannot be digging all those as the article seems to focus on , the focus should be on finding a resolution to Riemanns hypothesis, which may have been found now. There is published reference on a web site but we will not state it here out of respect for Wikipedia We can do it faster. -- Vinoo Cameron ( talk) 22:36, 1 August 2009 (UTC)
mwatkins
at Exter University, maths.ex.ac.uk
would probably be interested in your proof. Type it up, submit it to the arXiv, then shoot him a letter.
CRGreathouse (
t |
c)
16:27, 8 August 2009 (UTC)LOOK CHECK THIS PROOF , We at inverse 19 mathematics, can within minutes contruct a 30 decimal prime number , we have done that but are not allowed to put it here, BUT YOU DO THIS AS PROOF take any number no matter how big and you want to find if it is prime, you plus 180+prime like 23 each succesive times from eith that number you can develop infinite series and select many primes within minutes -- Vinoo Cameron ( talk) 19:45, 13 August 2009 (UTC) . Tell Dr MATTHEW WATKINS of Primes to look at this strange www.inverse19mathematics.com, we are on the verge proving Riemanns in the correct use of the Prime numbers.-- Vinoo Cameron ( talk) 17:14, 8 August 2009 (UTC)-- Vinoo Cameron ( talk) 17:14, 8 August 2009 (UTC)
Number 49 reference here was a superlative slip so do not glee ove it please , but I did answer this "Big BerthaPrime syndrome below in a wrong segment by a superlative slip if you do not mind , BUT we also proved that there are only 72 primes that are base to the circle and the midline of a circle is girded by the primes at -1 and +1 2, 179,359, 181.and we used the two 1- primes 179 and 359 add 180 to each and produce a sertries , you will see for yourself that the Set of primes ends at 359 and 179 exactly , see the math do it yourself and see what you find in the series, and yes We could give you 5 numbers and in trillions we can predict that onc is a Prime for sure. soon we will be 100 percent accvurate , within 3 moths if I could get the alog companies to run a seies on all 72 primes sans 49 of course . I have formally asked alogarithm companies to run a series on 1-72 primes and find the alogaritm to primes like the genome project , Riemanns is passe. Primes eflect to 5/19 in a cicle I.E division of 360 , then what the hell are we loooking for Big Betha when the truth is always inversE ( did not the human genome teach us that . "Hey We Found the biggest Bertha Prime" to which the gods smile and say that "Size does not make mathamatics , nor a mans grace forb that matter-- -- Vinoo Cameron ( talk) 18:44, 13 August 2009 (UTC)
Which is what I meant to say when I cut "and the word prime is also used as an adjective". In "prime number", that's exactly what it is. The excised phrase was spectacularly uninformative. 63.249.96.218 ( talk) 09:00, 12 August 2009 (UTC)
I agree with that edit. Sentences like "This number is prime" make it perfectly obvious that it's an adjective. "Prime" means "not admitting a non-trivial factorization". It applies not only to numbers but also to polynomials and some other things. Michael Hardy ( talk) 18:17, 12 August 2009 (UTC)
...and so does everyone else, of course. When it's used in that sense, then it's a noun. But it's obviously also used as an adjective. There's no need for an explicit statement that it's also used as an adjective when that's staring everyone in the face. Michael Hardy ( talk) 23:14, 13 August 2009 (UTC)
To quote "any nonzero natural number n can be factored into primes" Just like to point out that the number 1 is a "nonzero natural number". FlipC ( talk) 14:01, 13 August 2009 (UTC)
About the first sentence:
a prime number (or a prime) is a natural number which has exactly two distinct natural number
Discussing with (non-scientist) friends on a forum, several of them commented that they found the wording "exactly two" confusing and suggested "two and only two". To me it's not confusing at all, but if someone finds it this way and the edit is trivial and not disrupting, I'd go for it. I tried to go but it got reverted. Any opinion? Thanks! -- Cyclopia ( talk) 18:47, 24 August 2009 (UTC)
Under the heading "Generating prime numbers", the text currently has a statement ending in -- the methods mentioned above under “Finding prime numbers”. However, there is no "Finding prime numbers" mentioned above, so this is a dangling reference.
I don't feel qualified to fix this, perhaps someone else is.
Esb ( talk) 21:12, 24 August 2009 (UTC)
Euclid Proved that there are infinitely primes.
Hello. I don't understand why a mention of Uncle Petros and Goldbach's Conjecture was removed with the comment "That MacGuffin shouldn't extend down to the level of what it actually means." -- in fact I don't understand that comment at all. Surely a novel with Goldbach's Conjecture in the title is suitable for the topic of Prime numbers in the arts and literature? What did I do wrong here? Wonky the Worm ( talk) 17:14, 7 November 2009 (UTC)
The main article could be improved by linking it up with an article on pseudo-random number generators.
For any large prime number, followed by the primes that come right after it, we naturally know ahead of time that the right-most bit of the integer is a '1' and the leftmost bit of the integer is a '1' but there ought to be some discussion as to the distribution of ones and zeroes between those two bits, and whether, by concatenating these series together, you get any reasonably random distribution of zeroes and ones. Dexter Nextnumber ( talk) 03:44, 20 December 2009 (UTC)
I think the main article would be improved if there were a discussion of the internal structure of prime numbers. There is probably a huge amount of stuff on this sort of thing. I'm not exactly sure what I meant by plotting out the numbers, but clearly there are lots of ways of plotting numbers. Maybe use a cartesian grid, or in a spiral like what Ulam did. The numbers I am talking about, are in red. This sort of looks like a ton of overhead, just to get a single bit of randomness. On the other hand, I doubt very much it is random. These numbers probably observe a rule of generation that reveals itself just like any other series of numbers do. Dexter Nextnumber ( talk) 07:01, 21 December 2009 (UTC)
"There is no known formula yielding all primes and no composites..."
Sorry if I'm misunderstanding you, but...how about f(x) = 17. 4 T C 06:57, 8 January 2010 (UTC)
The opening lines refer to natural number, then go on to define non-primacy of one but not of zero. Peano's axiom #1 states zero is a natural number and this is also the case on the Wiki page for natural number which gives two conflicting definitions.
A way round would be to accept Peano but redefine an alternative set lacking zero, using the traditional title of counting numbers.
A better work-around in my view would be to define prime numbers through the impossibility of their arising through multiplication. A specific for this particular multiplication should refer to any pair of natural numbers which were generated earlier in sequence by Peano's (axiomatic) successor function.
This automatically excludes zero and one from being prime. —Preceding unsigned comment added by 84.228.30.202 ( talk) 08:35, 21 January 2010 (UTC)
The article does not mention a significant bit of information, namely that the ratio of the number of prime numbers to the number of natural numbers is zero, meaning that if one defines the function P(N) as the number of different primes less than N (so for example P(100)=25) then P(N)/N -> 0 when N -> infinity. I added this bit of information in the article, but user JohnBlackburne removed it saying it was "meaningless". Perhaps the wording I chose ("the ratio of prime numbers to natural numbers is zero") was not clear. If anybody knows a more concise way to express it, please go ahead and add it to the article. Dianelos ( talk) 00:05, 22 February 2010 (UTC)
An anonymous editor has twice added to the list of prime numbers in recent hours. Maybe we should include a good explanation of why 1 is not so considered. Here's how I think of it. Say I look at a positive integer such as 126. I can split it thus:
Then I can split 6 and 21 further, thus:
At this point the only way to split it further is to break off 1s:
and that can continue forever, but it doesn't reduce the 2, the two 3s, or the 7 to smaller numbers, and splitting off 1s gives no information about the number you started with, since going on forever splitting off 1s looks exactly the same regardless of what number you start with.
Thus we have three categories of numbers: composite numbers like 126, which we can split; prime numbers like 2, 3, and 7, which we cannot split except by breaking off 1s while not reducing the numbers we're trying to split; and the one remaining number: 1. Michael Hardy ( talk) 21:38, 2 March 2010 (UTC)
This article ought to mention that the density of primes is 0, i.e. the proportion of numbers less than n that are prime can be made as close as desired to 0 by making n big enough. Maybe a short proof could be included too. Or maybe one could link to a separate article about that result. Michael Hardy ( talk) 21:57, 20 March 2010 (UTC)
It certainly has much easier proofs than that of the prime number theorem. Some are very short and elementary—just a paragraph. Michael Hardy ( talk) 01:24, 21 March 2010 (UTC)
What is the point of the new graph File:Dp txt2.png, which has been added to the Gaps between primes section ? As far as I can see, it plots the nth prime against the difference between squares of consecutive primes . Since , you would expect the points on the grpah to lie along the lines with slopes 1/4, 1/8, 1/12 etc. - which is exactly what the graph shows. This seems to be (a) trivial and (b) not an illustration of anything in the article text. Unless someone can explain the point of this graph, I will remove it. Gandalf61 ( talk) 09:14, 28 March 2010 (UTC)
Is there any formula that computes (approximately, of course) the n-th prime number? For example, this formula would give f(1) = 2, f(2) = 3, f(3) = 5 (approximately). Albmont ( talk) 19:49, 14 April 2010 (UTC)
According to some web sites, the sequence 1,3,5,7,11,13,17,19... (with 1 and without 2) is considered the prime numbers for the purpose of something related to the Big Bang. Any references to this in Wikipedia?? Georgia guy ( talk) 15:07, 4 May 2010 (UTC)
As current article suggests,for carring out trial division for given number N it is necessary to divide it by all integer number less than its squre root. but altenatively it is possible to dividide N only by the prime numbers less than its squre root. To illustrate the reason, notice an example of trial division for number 97. As suggested in the current article, we should divide 97 by 1,2,3,4,5,6,7,8,9. but it is easy to undrestand if 97 fails to be a multiple of 2, it can not be a multiple of 4,6 and 8. in a same way because 97 is not multiple of 3 it is not multiple of 6 and 9. thus, we can result that only prime numbers less than or equal to the given number shold be involved in trial division. In brief it is sugested to edit the according part of the article —Preceding unsigned comment added by Babahadi ( talk • contribs) 13:39, 2 June 2010 (UTC)
I completely got the poin you did mention. I think you are right. But is it not possible to edit text and add the method I did mention as a easier or less time-consuming approch? 94.74.158.16 ( talk) 03:31, 3 June 2010 (UTC)
Ranjitr303 ( talk) 07:07, 24 June 2010 (UTC)can somebody tell me a formula for finding n succesive numbers such that within the n succesive numbers none is a prime. i had read about this in some book but i am not sure whether it is related to the great Ramanujan ?
It is claimed that there is an equation
(- when primes of the form 4k+1,and + when primes of the form 4k+3)
which leads to the irrationality of π implying the infinitude of primes. That equation does not seem to be sourced; even if it were, the equation:
provides a much simpler proof from the transcendence of π to the infinitude of primes. — Arthur Rubin (talk) 18:21, 14 September 2010 (UTC)
I never understood why the number 1 is not a prime number.
I always get "The number 1 is by definition not a prime number. "
But why? Why did they decide to not make 1 a prime number. It would seem so natural and complete to allow the number 1 to be a prime number. —Preceding unsigned comment added by 208.251.83.66 ( talk) 23:43, 27 September 2010 (UTC)
This recent addit (re)added some content about "generalized primes". It is clear that the content should be shrunk to at most one sentence, otherwise giving undue weight to this single paper. The question is: should we have this in the article at all? The paper, "International Journal of Algebra" does appear in MathSciNet (does this mean it is peer reviewed?), but is a young, and looking at its citation quotient, not very respected journal. Which rises the question, whether this article should mention that paper. Jakob.scholbach ( talk) 23:31, 5 January 2011 (UTC)
I reverted Cyclopia's recent revert of my attempts to make this a better article. Here is why: the proof should point out clearly that the fundamental th. of ar. is used. This was somewhat disguised previously. Of course, this is still far from perfect, especially what concerns wording etc., but do I think it is a step in the right direction. Also, I did not remove the Green-Tao, theorem, I just moved it to a more appropriate place. Jakob.scholbach ( talk) 21:59, 27 January 2011 (UTC)
As for applications of prime numbers, this mathoverflow thread contains some info. The book of Pomerance and Crandall mentions random number generators, quasi Monte Carlo numerical integration and cryptographical applications.
Does anyone have other applications (ideally with sources?). Jakob.scholbach ( talk) 22:45, 31 January 2011 (UTC)
regular prime, RSA number, supersingular prime, Adleman–Pomerance–Rumely primality test, Giuga's conjecture, Pepin's test, Prime k-tuple, k-tuple conjecture, number field sieve, Chen's theorem, Fermat's theorem on sums of two squares, Mills' theorem, Schnirelmann's theorem, smooth number, Elliptic curve factorization
Jakob.scholbach ( talk) 23:06, 31 January 2011 (UTC)
I did some edits to the math formatting of the first few sections of the article this morning, but from comments on talk Jakob does not seem to think they were an improvement. As I see it, we have three options for math formatting, and should choose one of them and follow it as consistently as possible. The options are:
My personal preference is the {{ math}} template, because I think the serif font makes the math stand out a little from the body text (so that you know to read it differently) and because (unlike the default sans-serif) it's possible to distinguish a capital i, lowercase L, and vertical bar |: compare sans-serif I l | vs serif I l |. However, Jakob seems to be of the opinion that the math should blend as much as possible with the text and therefore that we should use the usual sans-serif font. Another potential issue with {{ math}} is that it also uses the serif font for digits, so for best consistency of formatting it would need to be used for formulas containing only digits (of which we have many) as well as formulas containing variables (fewer). I think we are both agreed that bitmaps are ugly (but that they may be unavoidable for some complicated formulae). There seems to be no general agreement on which of these options should be used on Wikipedia more broadly (see Wikipedia talk:Manual of Style (mathematics)), but maybe we can at least come to something resembling a consensus for this one page. Does anyone else care to weigh in on this issue? — David Eppstein ( talk) 00:10, 28 January 2011 (UTC)
This was an extraordinarily bad edit. That Euclid's proof was by contradiction is false and is unfair to Euclid. It is true that quite a few respectable mathematicians assert this. Dirichlet was one of those. G. H. Hardy was another, although he changed his view on this, I suspect under the influence of his co-author Wright. That proves that mathematicians aren't really all that good at history. And maybe most historians aren't so good at mathematics, so they don't work on this either. My joint paper with Catherine Woodgold demolishes the myth and also shows why the proof by contradiction is inferior to the one that Euclid wrote. I've cited it in the article. Michael Hardy ( talk) 02:46, 29 January 2011 (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 4 | Archive 5 | Archive 6 | Archive 7 | Archive 8 | Archive 9 |
Some page format added. Charles Matthews 07:49, 4 May 2004 (UTC)
It's me the only one (screen resolution: 1024 x 768 in a 15'' LCD) that, in this image added on August the 6th [1], can see nothing more than a reddish black rectangle? Maybe the color choice is great for a background, but no to illustrate this article. Maybe in black and white it could be used here, otherwise, I think it would be better to remove it. -- 79.150.126.120 ( talk) 13:31, 26 August 2008 (UTC)
I second that. Apart from the poor colour choice, it is patently useless to display a small thumb of a huge image in which every pixel is supposed to carry one bit of information. It results in a random meaningless blur. — Emil J. (formerly EJ) 14:01, 26 August 2008 (UTC)
All I see is a black rectangle. The other problem (but easily fixed) is that it's uncouth to use an asterisk for ordinary multiplication, as in 3 + (1281 * 1024 * 2). One should write 3 + (1281 × 1024 × 2). Michael Hardy ( talk) 18:14, 12 August 2009 (UTC)
There's little more besides a headline stating more information to come soon, but http://mersenne.org/prime.htm claims to possibly have found the 45th mersenne prime number. Slashdot has also covered it. -- 76.85.144.126 ( talk) 00:28, 28 August 2008 (UTC)
FYI: Now there is news of a 46th known Mersenne prime see Fox News reported Sept 27, 2008 —Preceding unsigned comment added by 203.171.45.66 ( talk) 21:31, 27 September 2008 (UTC)
I've removed some material beginning Chris Curtis, a computer scientist from Auckland New Zealand showed the first predictive method for prime numbers by showing the first complete pattern in a method also used on the same day to rationalise Pi. as possible WP:OR, WP:PEACOCK terms, unsourced and contentious. Richard Pinch ( talk) 11:27, 20 September 2008 (UTC)
Should Wilson's theorem be listed among the primality tests? I know it is inefficient, but it does show that a number can be tested for primality with only one division. Asmeurer ( talk ♬ contribs) 06:09, 20 November 2008 (UTC)
I have always been wondering whether prime numbers exist in nature some way or another. Take a look in this article at what Stephen Wolfram has got to say about the rings of Saturn (it is somewhere in the middle): http://www.kurzweilai.net/meme/frame.html?main=memelist.html?m=17%23646. Does anybody know more about that and can give some examples in the article? —Preceding unsigned comment added by 89.236.28.157 ( talk) 10:37, 31 December 2008 (UTC)
I put here a number of items that I may put to the see also section or the main text. I will undertake some revision and expansion of the text shortly. Jakob.scholbach ( talk) 19:59, 31 March 2009 (UTC)
The distinction between Prime and Irreducible is never made in this article. One reason for this is because the article addresses primality only in the natural numbers. Both of these issues should be addressed. For example in other rings such as , prime and irreducible are not equivalent as in . This is a very important distinction that is (as far as I can find) never made on Wikipedia.
If generalizing this article would make it too cluttered or lengthy, other articles about Prime and Irreducible should be created. —Preceding unsigned comment added by Silverhammermba ( talk • contribs) 15:40, 16 April 2009 (UTC)
Dear Sir:
I'm sorry, but the way that the phrase below, took from the text, was whritten, lead us to an ideia that Euclid cames after Mersenne.
"Euclid also showed how to construct a perfect number from a Mersenne prime".
Forgive me for any inconvenience.
189.106.2.214 ( talk) 02:26, 23 April 2009 (UTC)
I am curious as to why there are so many mentions of means to determine primality of numbers, when the classic means to find primes is not mentioned anywhere in the Prime Number article: the Sieve of Eratosthenes. The sieve is not even listed in the See Also section. William R. Buckley ( talk) 16:55, 24 May 2009 (UTC)
We have to painfully state that it is more important to know that prime lie on a line and predict that line which breakthrough may have already happened using inverse 19 , because there are infinite primes, we cannot be digging all those as the article seems to focus on , the focus should be on finding a resolution to Riemanns hypothesis, which may have been found now. There is published reference on a web site but we will not state it here out of respect for Wikipedia We can do it faster. -- Vinoo Cameron ( talk) 22:36, 1 August 2009 (UTC)
mwatkins
at Exter University, maths.ex.ac.uk
would probably be interested in your proof. Type it up, submit it to the arXiv, then shoot him a letter.
CRGreathouse (
t |
c)
16:27, 8 August 2009 (UTC)LOOK CHECK THIS PROOF , We at inverse 19 mathematics, can within minutes contruct a 30 decimal prime number , we have done that but are not allowed to put it here, BUT YOU DO THIS AS PROOF take any number no matter how big and you want to find if it is prime, you plus 180+prime like 23 each succesive times from eith that number you can develop infinite series and select many primes within minutes -- Vinoo Cameron ( talk) 19:45, 13 August 2009 (UTC) . Tell Dr MATTHEW WATKINS of Primes to look at this strange www.inverse19mathematics.com, we are on the verge proving Riemanns in the correct use of the Prime numbers.-- Vinoo Cameron ( talk) 17:14, 8 August 2009 (UTC)-- Vinoo Cameron ( talk) 17:14, 8 August 2009 (UTC)
Number 49 reference here was a superlative slip so do not glee ove it please , but I did answer this "Big BerthaPrime syndrome below in a wrong segment by a superlative slip if you do not mind , BUT we also proved that there are only 72 primes that are base to the circle and the midline of a circle is girded by the primes at -1 and +1 2, 179,359, 181.and we used the two 1- primes 179 and 359 add 180 to each and produce a sertries , you will see for yourself that the Set of primes ends at 359 and 179 exactly , see the math do it yourself and see what you find in the series, and yes We could give you 5 numbers and in trillions we can predict that onc is a Prime for sure. soon we will be 100 percent accvurate , within 3 moths if I could get the alog companies to run a seies on all 72 primes sans 49 of course . I have formally asked alogarithm companies to run a series on 1-72 primes and find the alogaritm to primes like the genome project , Riemanns is passe. Primes eflect to 5/19 in a cicle I.E division of 360 , then what the hell are we loooking for Big Betha when the truth is always inversE ( did not the human genome teach us that . "Hey We Found the biggest Bertha Prime" to which the gods smile and say that "Size does not make mathamatics , nor a mans grace forb that matter-- -- Vinoo Cameron ( talk) 18:44, 13 August 2009 (UTC)
Which is what I meant to say when I cut "and the word prime is also used as an adjective". In "prime number", that's exactly what it is. The excised phrase was spectacularly uninformative. 63.249.96.218 ( talk) 09:00, 12 August 2009 (UTC)
I agree with that edit. Sentences like "This number is prime" make it perfectly obvious that it's an adjective. "Prime" means "not admitting a non-trivial factorization". It applies not only to numbers but also to polynomials and some other things. Michael Hardy ( talk) 18:17, 12 August 2009 (UTC)
...and so does everyone else, of course. When it's used in that sense, then it's a noun. But it's obviously also used as an adjective. There's no need for an explicit statement that it's also used as an adjective when that's staring everyone in the face. Michael Hardy ( talk) 23:14, 13 August 2009 (UTC)
To quote "any nonzero natural number n can be factored into primes" Just like to point out that the number 1 is a "nonzero natural number". FlipC ( talk) 14:01, 13 August 2009 (UTC)
About the first sentence:
a prime number (or a prime) is a natural number which has exactly two distinct natural number
Discussing with (non-scientist) friends on a forum, several of them commented that they found the wording "exactly two" confusing and suggested "two and only two". To me it's not confusing at all, but if someone finds it this way and the edit is trivial and not disrupting, I'd go for it. I tried to go but it got reverted. Any opinion? Thanks! -- Cyclopia ( talk) 18:47, 24 August 2009 (UTC)
Under the heading "Generating prime numbers", the text currently has a statement ending in -- the methods mentioned above under “Finding prime numbers”. However, there is no "Finding prime numbers" mentioned above, so this is a dangling reference.
I don't feel qualified to fix this, perhaps someone else is.
Esb ( talk) 21:12, 24 August 2009 (UTC)
Euclid Proved that there are infinitely primes.
Hello. I don't understand why a mention of Uncle Petros and Goldbach's Conjecture was removed with the comment "That MacGuffin shouldn't extend down to the level of what it actually means." -- in fact I don't understand that comment at all. Surely a novel with Goldbach's Conjecture in the title is suitable for the topic of Prime numbers in the arts and literature? What did I do wrong here? Wonky the Worm ( talk) 17:14, 7 November 2009 (UTC)
The main article could be improved by linking it up with an article on pseudo-random number generators.
For any large prime number, followed by the primes that come right after it, we naturally know ahead of time that the right-most bit of the integer is a '1' and the leftmost bit of the integer is a '1' but there ought to be some discussion as to the distribution of ones and zeroes between those two bits, and whether, by concatenating these series together, you get any reasonably random distribution of zeroes and ones. Dexter Nextnumber ( talk) 03:44, 20 December 2009 (UTC)
I think the main article would be improved if there were a discussion of the internal structure of prime numbers. There is probably a huge amount of stuff on this sort of thing. I'm not exactly sure what I meant by plotting out the numbers, but clearly there are lots of ways of plotting numbers. Maybe use a cartesian grid, or in a spiral like what Ulam did. The numbers I am talking about, are in red. This sort of looks like a ton of overhead, just to get a single bit of randomness. On the other hand, I doubt very much it is random. These numbers probably observe a rule of generation that reveals itself just like any other series of numbers do. Dexter Nextnumber ( talk) 07:01, 21 December 2009 (UTC)
"There is no known formula yielding all primes and no composites..."
Sorry if I'm misunderstanding you, but...how about f(x) = 17. 4 T C 06:57, 8 January 2010 (UTC)
The opening lines refer to natural number, then go on to define non-primacy of one but not of zero. Peano's axiom #1 states zero is a natural number and this is also the case on the Wiki page for natural number which gives two conflicting definitions.
A way round would be to accept Peano but redefine an alternative set lacking zero, using the traditional title of counting numbers.
A better work-around in my view would be to define prime numbers through the impossibility of their arising through multiplication. A specific for this particular multiplication should refer to any pair of natural numbers which were generated earlier in sequence by Peano's (axiomatic) successor function.
This automatically excludes zero and one from being prime. —Preceding unsigned comment added by 84.228.30.202 ( talk) 08:35, 21 January 2010 (UTC)
The article does not mention a significant bit of information, namely that the ratio of the number of prime numbers to the number of natural numbers is zero, meaning that if one defines the function P(N) as the number of different primes less than N (so for example P(100)=25) then P(N)/N -> 0 when N -> infinity. I added this bit of information in the article, but user JohnBlackburne removed it saying it was "meaningless". Perhaps the wording I chose ("the ratio of prime numbers to natural numbers is zero") was not clear. If anybody knows a more concise way to express it, please go ahead and add it to the article. Dianelos ( talk) 00:05, 22 February 2010 (UTC)
An anonymous editor has twice added to the list of prime numbers in recent hours. Maybe we should include a good explanation of why 1 is not so considered. Here's how I think of it. Say I look at a positive integer such as 126. I can split it thus:
Then I can split 6 and 21 further, thus:
At this point the only way to split it further is to break off 1s:
and that can continue forever, but it doesn't reduce the 2, the two 3s, or the 7 to smaller numbers, and splitting off 1s gives no information about the number you started with, since going on forever splitting off 1s looks exactly the same regardless of what number you start with.
Thus we have three categories of numbers: composite numbers like 126, which we can split; prime numbers like 2, 3, and 7, which we cannot split except by breaking off 1s while not reducing the numbers we're trying to split; and the one remaining number: 1. Michael Hardy ( talk) 21:38, 2 March 2010 (UTC)
This article ought to mention that the density of primes is 0, i.e. the proportion of numbers less than n that are prime can be made as close as desired to 0 by making n big enough. Maybe a short proof could be included too. Or maybe one could link to a separate article about that result. Michael Hardy ( talk) 21:57, 20 March 2010 (UTC)
It certainly has much easier proofs than that of the prime number theorem. Some are very short and elementary—just a paragraph. Michael Hardy ( talk) 01:24, 21 March 2010 (UTC)
What is the point of the new graph File:Dp txt2.png, which has been added to the Gaps between primes section ? As far as I can see, it plots the nth prime against the difference between squares of consecutive primes . Since , you would expect the points on the grpah to lie along the lines with slopes 1/4, 1/8, 1/12 etc. - which is exactly what the graph shows. This seems to be (a) trivial and (b) not an illustration of anything in the article text. Unless someone can explain the point of this graph, I will remove it. Gandalf61 ( talk) 09:14, 28 March 2010 (UTC)
Is there any formula that computes (approximately, of course) the n-th prime number? For example, this formula would give f(1) = 2, f(2) = 3, f(3) = 5 (approximately). Albmont ( talk) 19:49, 14 April 2010 (UTC)
According to some web sites, the sequence 1,3,5,7,11,13,17,19... (with 1 and without 2) is considered the prime numbers for the purpose of something related to the Big Bang. Any references to this in Wikipedia?? Georgia guy ( talk) 15:07, 4 May 2010 (UTC)
As current article suggests,for carring out trial division for given number N it is necessary to divide it by all integer number less than its squre root. but altenatively it is possible to dividide N only by the prime numbers less than its squre root. To illustrate the reason, notice an example of trial division for number 97. As suggested in the current article, we should divide 97 by 1,2,3,4,5,6,7,8,9. but it is easy to undrestand if 97 fails to be a multiple of 2, it can not be a multiple of 4,6 and 8. in a same way because 97 is not multiple of 3 it is not multiple of 6 and 9. thus, we can result that only prime numbers less than or equal to the given number shold be involved in trial division. In brief it is sugested to edit the according part of the article —Preceding unsigned comment added by Babahadi ( talk • contribs) 13:39, 2 June 2010 (UTC)
I completely got the poin you did mention. I think you are right. But is it not possible to edit text and add the method I did mention as a easier or less time-consuming approch? 94.74.158.16 ( talk) 03:31, 3 June 2010 (UTC)
Ranjitr303 ( talk) 07:07, 24 June 2010 (UTC)can somebody tell me a formula for finding n succesive numbers such that within the n succesive numbers none is a prime. i had read about this in some book but i am not sure whether it is related to the great Ramanujan ?
It is claimed that there is an equation
(- when primes of the form 4k+1,and + when primes of the form 4k+3)
which leads to the irrationality of π implying the infinitude of primes. That equation does not seem to be sourced; even if it were, the equation:
provides a much simpler proof from the transcendence of π to the infinitude of primes. — Arthur Rubin (talk) 18:21, 14 September 2010 (UTC)
I never understood why the number 1 is not a prime number.
I always get "The number 1 is by definition not a prime number. "
But why? Why did they decide to not make 1 a prime number. It would seem so natural and complete to allow the number 1 to be a prime number. —Preceding unsigned comment added by 208.251.83.66 ( talk) 23:43, 27 September 2010 (UTC)
This recent addit (re)added some content about "generalized primes". It is clear that the content should be shrunk to at most one sentence, otherwise giving undue weight to this single paper. The question is: should we have this in the article at all? The paper, "International Journal of Algebra" does appear in MathSciNet (does this mean it is peer reviewed?), but is a young, and looking at its citation quotient, not very respected journal. Which rises the question, whether this article should mention that paper. Jakob.scholbach ( talk) 23:31, 5 January 2011 (UTC)
I reverted Cyclopia's recent revert of my attempts to make this a better article. Here is why: the proof should point out clearly that the fundamental th. of ar. is used. This was somewhat disguised previously. Of course, this is still far from perfect, especially what concerns wording etc., but do I think it is a step in the right direction. Also, I did not remove the Green-Tao, theorem, I just moved it to a more appropriate place. Jakob.scholbach ( talk) 21:59, 27 January 2011 (UTC)
As for applications of prime numbers, this mathoverflow thread contains some info. The book of Pomerance and Crandall mentions random number generators, quasi Monte Carlo numerical integration and cryptographical applications.
Does anyone have other applications (ideally with sources?). Jakob.scholbach ( talk) 22:45, 31 January 2011 (UTC)
regular prime, RSA number, supersingular prime, Adleman–Pomerance–Rumely primality test, Giuga's conjecture, Pepin's test, Prime k-tuple, k-tuple conjecture, number field sieve, Chen's theorem, Fermat's theorem on sums of two squares, Mills' theorem, Schnirelmann's theorem, smooth number, Elliptic curve factorization
Jakob.scholbach ( talk) 23:06, 31 January 2011 (UTC)
I did some edits to the math formatting of the first few sections of the article this morning, but from comments on talk Jakob does not seem to think they were an improvement. As I see it, we have three options for math formatting, and should choose one of them and follow it as consistently as possible. The options are:
My personal preference is the {{ math}} template, because I think the serif font makes the math stand out a little from the body text (so that you know to read it differently) and because (unlike the default sans-serif) it's possible to distinguish a capital i, lowercase L, and vertical bar |: compare sans-serif I l | vs serif I l |. However, Jakob seems to be of the opinion that the math should blend as much as possible with the text and therefore that we should use the usual sans-serif font. Another potential issue with {{ math}} is that it also uses the serif font for digits, so for best consistency of formatting it would need to be used for formulas containing only digits (of which we have many) as well as formulas containing variables (fewer). I think we are both agreed that bitmaps are ugly (but that they may be unavoidable for some complicated formulae). There seems to be no general agreement on which of these options should be used on Wikipedia more broadly (see Wikipedia talk:Manual of Style (mathematics)), but maybe we can at least come to something resembling a consensus for this one page. Does anyone else care to weigh in on this issue? — David Eppstein ( talk) 00:10, 28 January 2011 (UTC)
This was an extraordinarily bad edit. That Euclid's proof was by contradiction is false and is unfair to Euclid. It is true that quite a few respectable mathematicians assert this. Dirichlet was one of those. G. H. Hardy was another, although he changed his view on this, I suspect under the influence of his co-author Wright. That proves that mathematicians aren't really all that good at history. And maybe most historians aren't so good at mathematics, so they don't work on this either. My joint paper with Catherine Woodgold demolishes the myth and also shows why the proof by contradiction is inferior to the one that Euclid wrote. I've cited it in the article. Michael Hardy ( talk) 02:46, 29 January 2011 (UTC)