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Some page format added. [[User:Charles Matthews|Charles Matthews]] 07:49, 4 May 2004 (UTC)

== Black image ==

[[Image:Primes 1281x1024.PNG|thumb|320px|right|The distribution of all odd numbers for prime numbers in the range of 3 to 2,623,491 (3 + (1281 * 1024 * 2)), from left to right and top to bottom, where each pixel represents a number. Red pixels mean that number is prime and Black means it is not prime.]]
It's me the only one (screen resolution: 1024 x 768 in a 15<nowiki>''</nowiki> LCD) that, in this image added on August the 6th [http://en.wikipedia.org/enwiki/w/index.php?title=Prime_number&diff=230262578&oldid=230182291], 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. --[[Special:Contributions/79.150.126.120|79.150.126.120]] ([[User talk: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.  — [[User:EmilJ|Emil]] [[User talk:EmilJ|J.]] <span style="font-size:80%">(formerly EJ)</span> 14:01, 26 August 2008 (UTC)

:Thirded.  [[User:Oli Filth|Oli Filth]]<sup>([[User talk:Oli Filth|talk]]|[[Special:Contributions/Oli_Filth|contribs]])</sup> 22:57, 26 August 2008 (UTC)
::Oli Filth has removed it and I support that. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 21:43, 27 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 {{nowrap|3 + (1281 * 1024 * 2)}}.  One should write&nbsp;{{nowrap|3 + (1281 &times; 1024 &times; 2)}}. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 18:14, 12 August 2009 (UTC)

== New record prime - 45th mersenne prime? ==

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. [http://science.slashdot.org/science/08/08/27/2220226.shtml Slashdot] has also covered it. --[[Special:Contributions/76.85.144.126|76.85.144.126]] ([[User talk:76.85.144.126|talk]]) 00:28, 28 August 2008 (UTC)

:It has been at the end of [[Mersenne prime#List of known Mersenne primes]] for 4 days.[http://en.wikipedia.org/enwiki/w/index.php?title=Mersenne_prime&diff=233958929&oldid=232661889] I don't think we should start updating many other articles (like [[Prime number]], [[Largest known prime]], [[Great Internet Mersenne Prime Search]], ...) with speculation if it's still unverified and of unknown size. Wikipedia is not a news site and can wait for a verification. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 01:03, 28 August 2008 (UTC)

::I agree. GIMPS says the verification runs might be complete by September 16. We can wait until then. [[User:Anton Mravcek|Anton Mravcek]] ([[User talk:Anton Mravcek|talk]]) 21:03, 28 August 2008 (UTC)

== 46th mersenne prime ==

FYI: Now there is news of a 46th known Mersenne prime see [http://www.foxnews.com/story/0,2933,429201,00.html Fox News] reported Sept 27, 2008  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/203.171.45.66|203.171.45.66]] ([[User talk:203.171.45.66|talk]]) 21:31, 27 September 2008 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:This Mersenne prime is already mentioned at [[Prime number#Location of the largest known prime]]. It was the 45th to be discovered and is currently the largest known and 46th in order of size. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 23:13, 27 September 2008 (UTC)

== Theory of everything ==

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.  [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 11:27, 20 September 2008 (UTC)

== Wilson's theorem and primality tests ==

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.  <span style="color:#008888;">[[User:Asmeurer|<span style="color:#008888;">Asmeurer</span>]] ([[User talk:Asmeurer|<span style="color:#008888;">talk</span>]]  ♬ [[Special:Contributions/Asmeurer|<span style="color:#008888;">contribs</span>]])</span> 06:09, 20 November 2008 (UTC)

:Primality can be tested without ''any'' division using Eratosthenes' sieve, which is still more efficient than Wilson's theorem. Nothing remarkable about that. —&nbsp;[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 10:44, 20 November 2008 (UTC)

:The method with a single division at the end is really, really bad - it uses far more time ''and'' space than any other method (space is &Theta;((log ''n'')<sup>''n''</sup>) bits, superexponential growth). If you're going to use Wilson's theorem you'd really want to reduce mod ''p'' at each step. In any case I've added Wilson's theorem at [[primality test]], and don't think it deserves a mention here. [[User:Dcoetzee|Dcoetzee]] 19:39, 14 January 2009 (UTC)

::I agree (except even slower methods can be constructed and some people have). It's useless as a computational primality test but it's a notable and sometimes useful property of prime numbers with a deserved mention in [[Prime number#Properties]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 23:00, 14 January 2009 (UTC)

== Occurrence of primes in nature ==

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?  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/89.236.28.157|89.236.28.157]] ([[User talk:89.236.28.157|talk]]) 10:37, 31 December 2008 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:Check out [[Cicada#Life cycle]] for one occurrence of primes in natures. [[User:RobHar|RobHar]] ([[User talk:RobHar|talk]]) 18:34, 31 December 2008 (UTC)

:The whole paragraph about starfish is irrelevant.  It provides no information about primes; this paragraph is about starfish and goes into far too much detail about starfish.  This article certainly shouldn't include a list of things in nature that are prime, and it also shouldn't include a list of things in nature that aren't prime, which is effectively the whole amount of information in this paragraph relating to primes.  Pointing out that the belief that prime numbers appear in nature [i]because[/i] they are prime is erroneous is fine, but going into a detailed discussion about starfish isn't necessary.  I'm going to remove the paragraph about starfish. -- D. A.  19 January 2009  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/129.97.185.230|129.97.185.230]] ([[User talk:129.97.185.230|talk]]) 21:46, 19 January 2009 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Possible further see also items ==

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. [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 19:59, 31 March 2009 (UTC)

* [[ISBN]]
* [[Eisenstein's_criterion]]
* [[Paillier_cryptosystem]]
* [[Prime ring]] / [[prime field]]
* [[Chebotarev's density theorem]]
* [[Ramanujan–Petersson conjecture]]
* [[Quadratic residuosity problem]]
* [[Chen's theorem]]
* [[prime power]]
* [[Burnside theorem]]
* [[multiplicative number theory]]
* [[large sieve]]
* [[Eisenstein prime]]
* [[Generating primes]]
* [[Artin–Schreier theory]]
* [[Cauchy's theorem (group theory)]]
* [[Mathematics in medieval Islam]]
* [[Ostrowski's theorem]]
* [[Elliptic curve primality proving]]
* [[Primes in arithmetic progression]]

== Distinction between Prime and Irreducible ==

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 <math>a+b\sqrt{D}</math>, prime and irreducible are not equivalent as in <math>\mathbb{N}</math>. 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.  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Silverhammermba|Silverhammermba]] ([[User talk:Silverhammermba|talk]] • [[Special:Contributions/Silverhammermba|contribs]]) 15:40, 16 April 2009 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

:A search for "irreducible" on the [[Prime number]] page immediately finds [[Prime number#Prime_elements_in_rings]] which begins:
:{{Main|Prime element|Irreducible element}}
:and then continues with a brief discussion of precisely the issue you bring up. —[[User:Dominus|Dominus]] ([[User talk:Dominus|talk]]) 16:02, 16 April 2009 (UTC)

== History - Euclid's x Mersenne ==

 
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.

[[Special:Contributions/189.106.2.214|189.106.2.214]] ([[User talk:189.106.2.214|talk]]) 02:26, 23 April 2009 (UTC)

== Eratosthenes ==

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.  [[User:William R. Buckley|William R. Buckley]] ([[User talk:William R. Buckley|talk]]) 16:55, 24 May 2009 (UTC)

:That sieve is mentioned in two different places, once in the history section (including a large graphic), and again in the section "verifying primality"  when it is described how to generate primes up to a certain number.  Note also that determining the primality of a number is a different (although related) problem from generating primes.  To determine the primality of a very large number by sieving all the numbers up to it would be ridiculously time-consuming.  --[[User:C S|C S]] ([[User talk:C S|talk]]) 17:32, 24 May 2009 (UTC)

::The title of this article is Prime Number, not Primality, nor Determination of Primality; perhaps my initial question was not well specified.  My concern is that the sieve is not mentioned at all in an article that by name purports to be about prime numbers.  At the very least, readers would expect a link from the article Prime Number to the article Sieve of Eratosthenes; this point is my main concern.  Also, there are means to determine primality of very large numbers using variations of the sieve, combined with Rabin's Witness to Primality; the key is finding good candidates.  [[User:William R. Buckley|William R. Buckley]] ([[User talk:William R. Buckley|talk]]) 05:04, 25 May 2009 (UTC)

:::Regarding "the sieve is not mentioned at all", please see my comment above: "That sieve is mentioned in two different places, once in the history section (including a large graphic), and again in the section "verifying primality"  when it is described how to generate primes up to a certain number."  Do you understand this?  '''The Sieve of Eratosthenes is linked more than once from this article, and there is a large graphic demonstrating the sieve'''.  

:::The primality test coverage is also only a small portion of a much larger article.  I don't understanding your beef with it.  --[[User:C S|C S]] ([[User talk:C S|talk]]) 05:21, 25 May 2009 (UTC)

::::Sorry, my mistake.  You are correct that the mechanism is discussed.  Thank you for pointing the err of my reading.  [[User:William R. Buckley|William R. Buckley]] ([[User talk:William R. Buckley|talk]]) 04:57, 26 May 2009 (UTC)
== PRIME NUMBERS WIKEPEDIA ==

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. --[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 22:36, 1 August 2009 (UTC)
: Considering the vast number of papers appearing constantly in hundreds of mathematics journals, reporting torrents of new discoveries in mathematics, I hardly think we're in a "morass", and I doubt you can do it faster.  It's already moving faster than at any time in history. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 16:31, 31 July 2009 (UTC)
:I saw there was a proof of the [[Riemann hypothesis]] announced some years ago by Archimedes Plutonium using [[p-adic numbers]], however most mathematicians have been unable to understand it so it hasn't been generally accepted. Hopefully your proof is easier to check as until it is accepted for publication by a reputable journal and other people write about it wikipedia can't do anything about it, it doesn't publish [[WP:OR|original research]]. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 22:58, 1 August 2009 (UTC)

:: Dr. Matthew Watkins (<code>mwatkins</code> at Exter University, <code>maths.ex.ac.uk</code> would probably be interested in your proof. Type it up, submit it to the arXiv, then shoot him a letter. [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 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 --[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk: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.--[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 17:14, 8 August 2009 (UTC)--[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 17:14, 8 August 2009 (UTC)

:::<small>Somehow I'm put in mind of the short story 'The Red Brain' by [[Donald Wandrei]] when reading that above. Not sure of the connection but I thought it was a good story. Perhaps I better just stay from this. [http://www.donherron.com/collecting_donald_wandrei.html Collecting Donald Wandrei] has the plot. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 17:56, 8 August 2009 (UTC)</small>
::" Somehow we just posted the five biggest primes, bigger than the current biggest  and a trial method to confirm these at sci math and Physics/astronomy forums an hour ago atleast we threw the manna in the wind rather than gather the Primes into a folded skirt  , rather than take 1 year this will take 30 minutes with a good calculator . Indeed "Truth is stranger than fiction" and the euphemism of the above proximate comments does not escape me, but the mongoose  goes staight for the Cobras head, then waste its red brain on euphemisms--that was the trouble with Mongomery according to Eisehower. In any case the Prime article is very good , but already obsolete.--[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 00:24, 9 August 2009 (UTC)  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]] • [[Special:Contributions/Vinoo Cameron|contribs]]) 00:21, 9 August 2009 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

::: A 30-digit prime in minutes? You need to get much more impressive before people will pay attention. [[PARI/GP]] generated a 30-digit prime (555688433941833690722771369677) in microseconds and verified its primality in 4ms. [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 06:48, 10 August 2009 (UTC)

:::::::::You miss the  point here , literality is not the piont here , the point is that with the series simple 23 or most primes  +180 for the half circle as the primes are arranged, you can extend a series to trillions, and at present there is a 1 in 3 chance it is a prime , in three months we might be able to be exact --[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 18:50, 13 August 2009 (UTC)--[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 18:50, 13 August 2009 (UTC) . .

::::Give him a break, It sounds like he can extend his methods easily to generate bigger one than the [[Largest known prime number]] at nearly 13 million digits. 180000000000........0003 with any number of zeroes should be a prime if I'm reading what he says right, oops that's divisible by 3, ah well lets try 7, oh dear 187 is divisible by 11, it must be I don't understand his method properly. :) [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 08:14, 10 August 2009 (UTC)

:::::He gave a (reasonably clear) example [http://groups.google.com/group/sci.math/browse_thread/thread/c5caf5db85792a98/e5058964c01de337?show_docid=e5058964c01de337 here], which asserts 49 is prime. ''[[User:Hut 8.5|<b style="color:#FF0000;">Hut 8.5</b>]]'' 09:34, 10 August 2009 (UTC)

:::::: That raises another interesting point.  Under the RH there are 1.05901756822452865561220555018…{{e|12978181}} primes under M43112609 (plus or minus 2.12{{e|6489100}}), while his method gives at least 1.75{{e|12978186}} primes. So this isn't just a proof that P = NP, it's also a disproof of the Riemann hypothesis.  Of course it's also a disproof of [[Rosser's theorem]] and so makes ZFC (and so Z and probably GST) inconsistent… [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 22:09, 10 August 2009 (UTC)

:::::::Why make it so complicated? Since 49 is prime, Robinson arithmetic (and therefore Burgess' ST) is inconsistent. —&nbsp;[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 10:06, 11 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-- --[[User:Vinoo Cameron|Vinoo Cameron]] ([[User talk:Vinoo Cameron|talk]]) 18:44, 13 August 2009 (UTC)

== "Prime" is an adjective. ==

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. [[Special:Contributions/63.249.96.218|63.249.96.218]] ([[User talk:63.249.96.218|talk]]) 09:00, 12 August 2009 (UTC)

:"[[wikt:prime|prime]]" alone can be both a noun and an adjective. I view "prime number" together as one noun with a precisely defined meaning like "Fibonacci number" and not as the adjective "prime" applied to the noun "number". I don't agree with your edit [http://en.wikipedia.org/enwiki/w/index.php?title=Prime_number&diff=307525468&oldid=307341842]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 12:54, 12 August 2009 (UTC)

::Nor do I, for the same reason. Revert? [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 14:05, 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. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 18:17, 12 August 2009 (UTC)
:Colloquially, I often encounter mathematicians who use "prime" as short for "prime number" as in "97 is the largest prime less than 100". [[User:Dcoetzee|Dcoetzee]] 22:44, 13 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. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 23:14, 13 August 2009 (UTC)
:Yeah, I agree with that - no need to spell out this kind of thing when it's clear from usage. Generally adjective definitions are introduced in mathematics with wording something like: "An X with property Y is said to be ''frabulous''." [[User:Dcoetzee|Dcoetzee]] 23:17, 13 August 2009 (UTC)

== FTA error in second paragraph? ==

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". [[User:FlipC|FlipC]] ([[User talk:FlipC|talk]]) 14:01, 13 August 2009 (UTC)

:Yes. And it is also the empty product of primes, so there is no error. —&nbsp;[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 14:08, 13 August 2009 (UTC)

:Got you there :) Actually I can see others complaining about that too so I think I'll include a reference to [[empty product]] [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 16:12, 13 August 2009 (UTC)

== Minor rewording of introduction ==

About the first sentence:
<blockquote>
a '''prime number''' (or a '''prime''') is a [[natural number]] which has exactly two ''distinct'' natural number
</blockquote>

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! --[[User:Cyclopia|Cyclopia]] ([[User talk:Cyclopia|talk]]) 18:47, 24 August 2009 (UTC)

:Agree it is strange wording. Some of the rest of the introduction should just be removed like that there is no formula for primes. How about
:A prime number is a natural number for which the only the only smaller divisor is 1.
:The text goes on later to reinforce that 1 is not  prime number. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 19:56, 24 August 2009 (UTC)

== dangling reference to "Finding prime numbers" ==

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.

[[User:Esb|Esb]] ([[User talk:Esb|talk]]) 21:12, 24 August 2009 (UTC)

Euclid Proved that there are infinitely primes.

== In the arts and literature ==

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?  [[User:Wonky the Worm|Wonky the Worm]] ([[User talk:Wonky the Worm|talk]]) 17:14, 7 November 2009 (UTC)

:This is already mentioned in [[Goldbach's conjecture]] where it is more strictly relevant.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 17:23, 7 November 2009 (UTC)

:That film is not relevant to the topic of this article. The film mentions Goldbach's conjecture as a plot device (the [[MacGuffin]]) which could be replaced by of loads of others without changing the film. But even if it had used Goldbach's conjecture as an integral part of the plot the only place the connection is relevant to is the article [[Goldbach's conjecture]]. Mentioning it here is like saying because a film had a golden eagle in it and golden eagles are birds and lay eggs then the articles about birds and eggs should mention the film. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 17:30, 7 November 2009 (UTC)
== Generating pseudo-random numbers ==

The main article could be improved by linking it up with an article on [[pseudo-random number generator]]s.

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.   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 03:44, 20 December 2009 (UTC)

:It is briefly mentioned in the applications section, but do not hesitate adding more material. [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 10:08, 20 December 2009 (UTC)

::My question relates to prime numbers greater than 3.   In other words, prime numbers of 5 and more.  These primes are big enough that their leftmost bits and rightmost bits can be disregarded.   The bits that remain are what I am curious about.   I am inclined to call these internal bits the "internal makeup" or "constitution" of the prime.   Numbers 0, 1, and 2 are not big enough to be helpful.   The bigger the prime, the more helpful it is.

::The natural number 5 is a prime with 3 bits.  Its binary representation is %101 with the percent key being the standard character for denoting binary numbers.   The internal bit for 5 lies in the middle, a zero.  The very next prime number, 7, has the following binary representation:   %111, and  it has a  '1' in the middle.   If we ignore the bit at the far left, and the far right, we are left with the bits inbetween.   The issue, here, is whether running these bits together by concatenation (or any other process, like multiplication), leads to a conspicuous pattern.

::Someone should write an article about the internal composition of a prime number, and research proving it is independent from, and related to, what came before.   Intuition says, of course, there is a direct relationship.   But graphic representations of this relationship can be very persuasive.  [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 23:42, 20 December 2009 (UTC)

:::Ahem.  You've misinterpreted how prime numbers are used in PRNGs.  — [[User:Arthur Rubin|Arthur Rubin]]  [[User talk:Arthur Rubin|(talk)]] 23:59, 20 December 2009 (UTC)

::::Okay, but I would still like to see an article on the internal makeup or composition of a prime number.   It's pretty clear I am not talking about [[factorization]].   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 00:08, 21 December 2009 (UTC)

:::::Here is a table I put together.   


[[Image:Prime-numbers-internal-structure.gif]]

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's spiral|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.   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 07:01, 21 December 2009 (UTC)

:I appreciate your ideas, but we're here to write an encyclopedia article, not to do research in number theory. Please have a look at [[WP:NOR]]. If you think to have something new to say on prime numbers, publish it in an academic journal first, and then come back. --[[User:Cyclopia|<span style="color:green;">Cycl</span><big>o</big><span style="color:green;">pia</span>]][[User talk:Cyclopia|<span style="color:red;"><sup>talk</sup></span>]] 09:52, 21 December 2009 (UTC)

::I think this is not a case of doing original research, but of being [[bold]] enough to say something that's obvious, and then hope that a real mathematician (not me)  will come along and explain why it's so obvious.  And secondly, I seem to remember an old issue of [[Scientific American]] where someone surmised that any bit in the "second" position (Bit 1 as opposed to Bit 0) tended to ''depart'' from an expected 50-50 probability as the number grew larger.   I think the issue was in the late 1970s, and I don't have my Scientific American collection anymore, so I have no way of doing the "research" that is needed.   I will also be [[bold]] enough to say that most [[machine language]] programmers notice the way a carry flag does not propagate evenly across the [[seed]] of a [[pseudo-random number generator]] per iteration of permutation as the number gets bigger and bigger.   (A very large seed is often very useful in generating very large random numbers)  At least I noticed that sort of thing 10 years ago.   I don't see anything novel about this.  To illustrate this, take a random number generator that relies on a 16 bit seed, and then compare it to the same random number generator that relies on a 256 bit seed, and so on.  But somebody somewhere years ago should have already mentioned that somewhere along the line.

::And we haven't even touched on "recomposition" of numbers extracted from the insides of primes.

::And using the same scheme, how often is an "internal prime" found inside a prime?  This is not the same question as finding factors to a [[composite number]].    [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 20:59, 21 December 2009 (UTC)

:It is a fundamental principle of wikipedia that it does not publish [[WP:Original research]]. That includes anything but the most trivial computation like changing feet into metres. Anything has be be published and reviewed before it is included in maths articles in line with wikipedia's requirement for [[WP:reliable sources]]. You can ask questions about maths at the [[WP:RD/MA||Mathematics reference desk]]. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 22:37, 21 December 2009 (UTC)

::As I already said, there is nothing original or novel about this.   Maybe you should go to your [[college professor]] and ask him why he didn't bother to teach you about this.   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 01:28, 22 December 2009 (UTC)

:::This does not belong on Wikipedia.  As for specifics, the rightmost ''k'' bits are approximately uniformly distributed over the possibilities, per [[Dirichlet's theorem on arithmetic progressions#Distribution]].  The left ''k'' digits are distributed according to the same distribution as that of "random" numbers, in general.  I don't recall the Wikipedia article on someone's distribution, but it's here, and the same quasi-argument works.  — [[User:Arthur Rubin|Arthur Rubin]]  [[User talk:Arthur Rubin|(talk)]] 02:30, 22 December 2009 (UTC)

::::The illustration belongs on Wikipedia because it basically asks a question, rather than tries to answer it.   The main article does not answer the question, and could be improved if it did, perhaps in an equally graphic form.   Thanks for the pointer to Dirchlet's theory.   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 22:32, 22 December 2009 (UTC)

:::::The principles of Wikipedia are summarized at [[WP:Five pillars]]. This is an encyclopaedia. It does not ask questions. It just reports what other people have written about. There is also a reference desk at [[WP:RD/MA]] where you can ask questions. If all the books saif the earth was flat then even though you could easily see it is round and had gone round it and could point to the shadow on the moon wikipedia would only describe the earth as flat. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 00:12, 23 December 2009 (UTC)

::::::People regularly go to the Talk page to discuss how to improve the main article.  Questions regularly come up in the Talk page.  It is possible questions are never fully answered.   Nor even partially answered.   And sometimes the answer is no.   But shouldn't there be a warning ahead of time that a particular question is not to be asked?   The question submitted relates directly to the internal construction of Prime numbers.   This is a matter of [[bold|boldness]], something that makes Wikipedia more interesting (or at least the facts in the articles more  [[notable|noteworthy]]), even if you or I find them misleading or unacceptable.   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 04:40, 23 December 2009 (UTC)

:::::::If you can find a reliable source that provides a significant discussion of your question, then it may meet [[Wikipedia:Notability]] and be eligible for inclusion here. However until you do that I think the consensus of the other editors here is that your question is not appropriate for the article at the present time. Feel free to continue asking more questions, but remember that we (and we includes you!) are working on an encyclopedia, not on [[Wikipedia:Original research]]. [[User:Ozob|Ozob]] ([[User talk:Ozob|talk]]) 06:29, 23 December 2009 (UTC)

:::::::Questions like this should be asked and answered at the [[WP:RD/MA|reference desk]]. If you cannot get a satisfactory answer there then there are [[Internet forum|other forums]] still better suited to such discussions.--[[User:JohnBlackburne|JohnBlackburne]] ([[User talk:JohnBlackburne|talk]]) 06:37, 23 December 2009 (UTC)

::::::::I disagree with all three of you.   The Talk page is the very ''best'' place to submit questions of this nature (specifically, relating to the improvement of the main page for Prime numbers).   That you think the main page is currently satisfactory in its limited information, says something about an ulterior motive or two.   Especially if one of you already knows of plenty of citations relating to the matter at hand, and chooses not to state them, which is how I see it.    And as for newsgroups in Usenet, ''nothing'' stifles a discussion faster than someone who comes along and swamps a newsgroup with a bunch of binaries.    Even one binary - in this case, a .GIF file - is enough to put a damper on the discussion.    [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 21:30, 23 December 2009 (UTC)

:::::::::As others have explained, the content needs to be encyclopedic in it its format and presentation, and needs to be based on existing [[WP:RS|reliable sources]], not [[WP:OR|original research]].  If you believe reliable source exist but don't know where to find them, the [[WP:RD/MA|Reference Desk]] is the place to bring it up.  I don't think I just said anything that hasn't already been said by others, but for what it's worth...  [[User:Mwalimu59|mwalimu59]] ([[User talk:Mwalimu59|talk]]) 23:17, 23 December 2009 (UTC)

:The GIF image cited above looks like a problem from a school textbook. It appears to set questions for the reader, which is non-encyclopedic in style.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 21:34, 23 December 2009 (UTC)

::That is true, the .GIF image would be unsuitable for the ''main'' page, but it is perfectly suitable for the ''Talk'' page.   It relates to the improvement of the main page if somebody somewhere already knows of some cites to support it.   [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 21:46, 23 December 2009 (UTC)

:::A huge amount has been published about prime numbers but [[prime number]] should have a limited size and is the main article for [[:Category:Prime numbers]] which has more than 100 articles including subcategories. Primality is not a base dependent property and the binary representation of prime numbers is not considered important so it doesn't belong in the article even if somebody digs up a reference. My own site actually has a table [http://users.cybercity.dk/~dsl522332/math/congruent-primes.htm#mod4] which shows a consequence of bit 1 of consecutive relatively small prime numbers ''not'' being random, but changing more frequently than staying the same. The leftmost bits of consecutive prime numbers are obviously extremely far from being random. They are constant for increasingly long intervals of consecutive primes because the same holds for ''all'' consecutive integers. That is an elementary property of any positional number system. Your image numbers are so small that the constant effect is limited but it is why all 6 prime numbers from 67 to 89 start with a red 0. If you want to discuss in a usenet group then just make a plain text version of your image which doesn't require graphics. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 02:14, 24 December 2009 (UTC)

::::To generate a random (i.e., unpredictable) number, most people just multiply a very big integer (or an array of such integers) by a very large prime, and do that again and again, for the length of the document, doing the same thing the same way perhaps with minor modifications.   Another way of generating a random number is reordering the bits in some other way than multiplying (like executing a lot of XOR instructions).   Another way is by running some or all of the bits backward.   As you noted, the bits on the far left don't change much.   And the bigger the number, the less often they change.  If you run the bits backwards, it doesn't solve the problem of encryption (versus cracking), it just delays it.  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]] • [[Special:Contributions/Dexter Nextnumber|contribs]]) 22:03, 24 December 2009 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:::::[[List of random number generators]] gives a number of methods of generating random numbers, there's lots of them. By the way doing a lot of XOR instructions is no better than doing just one. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 22:12, 24 December 2009 (UTC)

::::::Yes, I agree.  There are lots of them.   By the way, how many times you perform an XOR instruction depends on what kind of microprocessor you are using.   Try encrypting a 60K text document using a [[MOS Technologies|6502]].   You'll find yourself executing EOR over and over again, all the way through the document.    [[User:Dexter Nextnumber|Dexter Nextnumber]] ([[User talk:Dexter Nextnumber|talk]]) 21:55, 25 December 2009 (UTC)

== Just pointing out ==

"There is no known formula yielding all primes and no composites..."

Sorry if I'm misunderstanding you, but...how about f(x) = 17. [[User:4|4]] <small>[[User talk:4|T]] [[Special:Contributions/4|C]]</small> 06:57, 8 January 2010 (UTC)

:Sorry if I'm misunderstanding you, but what does this actually mean? If the equation yields only one result then it is trivial, since there are infinitely many primes. What the article means is that no known formula will generate a range of numbers all guaranteed to be prime.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 08:54, 8 January 2010 (UTC)

:You have read "all primes" to mean "only prime numbers", whereas it is intended to mean "all of the prime numbers". I agree the wording is ambiguous, so I have changed it in the article. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 10:06, 8 January 2010 (UTC)
== Ambiguity in opening paragraphs ==

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.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/84.228.30.202|84.228.30.202]] ([[User talk:84.228.30.202|talk]]) 08:35, 21 January 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:Not sure why 0 needs special treatment. It is obviously not prime because it has many factorisations apart from 0x1 ... 0 = 0x2 = 0x3 = 0x4 etc. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 08:54, 21 January 2010 (UTC)

: Yes, as I read it there's no way to interpret it as saying zero could be prime, i.e. none of the definitions include zero whether or not the natural numbers does.--<small>[[User:JohnBlackburne|JohnBlackburne]]</small><sup>[[User_talk:JohnBlackburne|words]]</sup><sub style="margin-left:-5.0ex;">[[Special:Contributions/JohnBlackburne|deeds]]</sub> 09:21, 21 January 2010 (UTC)

:That's shows a problem of wikipedia articles being independent and assuming things about each other. The natural numbers have two definitions now, one including zero and the usual definition with primes is the non-zero one, primes were around long before zero. I guess now the article will have to refer to positive integers. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 11:19, 21 January 2010 (UTC)

== Ratio of prime numbers ==

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. [[User:Dianelos|Dianelos]] ([[User talk:Dianelos|talk]]) 00:05, 22 February 2010 (UTC)
: As you [http://en.wikipedia.org/enwiki/w/index.php?title=Prime_number&action=historysubmit&diff=345434077&oldid=345363379 wrote]:
:::but the [[ratio]] of prime numbers to natural numbers is 0.
:it did not make sense. Expanded in your comment above I see what you mean. But it's a trivial observation. See e.g. the [[prime-counting function]] π(x). This is known to tend to x/ln(x) as x → ∞. So π(x):x ≈ 1/ln(x):1 = 1:ln(x). This trivially tends to 0:1 as x → ∞. See also the [[prime number theorem]] and the 4th paragraph of this article.--<small>[[User:JohnBlackburne|JohnBlackburne]]</small><sup>[[User_talk:JohnBlackburne|words]]</sup><sub style="margin-left:-2.0ex;">[[Special:Contributions/JohnBlackburne|deeds]]</sub> 00:29, 22 February 2010 (UTC)
== Why 1 is not considered prime ==

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:
: <math> 126 = 6 \times 21. \, </math>
Then I can split 6 and 21 further, thus:
: <math>126 = 6 \times 21 = (2 \times 3) \times (2 \times 7). \, </math>
At this point the only way to split it further is to break off&nbsp;1s:
: <math> 126 = 6 \times 21 = (2 \times 3) \times (2 \times 7) = (1\times 2 \times 1 \times 3) \times (1 \times 3 \times 1 \times 7) \, </math>
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:&nbsp;1. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:38, 2 March 2010 (UTC)

::The '''Primality of one''' section gives an explanation of why 1 is not considered prime, although some of it is a bit technical. My own view is that 1 is not really prime, but "sort of" prime because it is an integer. It is rather like asking if 1 is an odd or even number, which does not apply to it.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 21:53, 2 March 2010 (UTC)

:::One is not at all prime, nor at all composite, and it's very odd. The reason why one looks unique among the integers is because it's almost the only example of a third kind of number, the ''units'': These are the numbers which have a multiplicative inverse which is the same kind of number (in this case, an integer). The only other unit in the integers is &minus;1. If you consider the larger set of rational numbers, then all the non-zero numbers are units. But there are intermediate examples, too. For instance, you can consider all the numbers of the form ''x''/2<sup>''k''</sup> for some integers ''x'' and ''k''; in other words, you allow only denominators which are powers of two. You can still add, subtract, and multiply these numbers. But now 2, 4, 8, ... are units, because their inverses 1/2, 1/4, 1/8, ... are also in the same set that we're considering. There are a lot of other intermediate examples, such as [[Gaussian integer]]s and [[Eisenstein integer]]s, which turn up everywhere in [[number theory]]. [[User:Ozob|Ozob]] ([[User talk:Ozob|talk]]) 01:40, 3 March 2010 (UTC)

:::1 is most definitely an odd number. Why shouldn't the even-odd distinction apply to it?—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 11:04, 3 March 2010 (UTC)

::::See [[Evenness of zero]] to see how much of a meal some people make of that! [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]])
:::::I got sort of used to the fact that some people make a meal of anything related to number ''zero''. But what's wrong with ''one''? What did it do to anybody? It's an honest to god natural number under any definition of natural number I've ever seen, and it is not divisible by 2.—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 12:20, 3 March 2010 (UTC)

::::::1 is clearly not divisible by 2 with an integer result. However, as with prime numbers, the unit property of 1 means that it does not fit comfortably within the odd/even debate. Also, according to [http://mathworld.wolfram.com/OddNumber.html this definition from Wolfram], an odd number leaves a remainder of 1 when divided by 2. 1 does not do this, but as ever, further debate is likely.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 13:28, 3 March 2010 (UTC)
:::::::Huh ? So just what would you say ''is'' the remainder when 1 is divided by 2, if it is not 1 ?? [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 13:34, 3 March 2010 (UTC)
:::::::And what does the unit property have to do with it? It's relevant for primality, but not for parity.—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 13:47, 3 March 2010 (UTC)
:1 is an [[odd number]] because it passes the standard test ''n = 2k + 1'' when k is zero. 1 is created as the result of multiplication by zero, which is why it has unusual status. Zero's status as a [[natural number]] is the subject of debate.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 14:40, 3 March 2010 (UTC)
::Admit it EMilJ, rhat's a sound mathematical argument :) Actually as [[Parity (mathematics)]] says the Greeks had the same problem with 1. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 15:04, 3 March 2010 (UTC)
:::That the Greeks may have had a problem is interesting, but the argument is still unsound. The condition Ianmacm quoted from the [[parity (mathematics)]] article reads in full that ''n'' = 2''k'' + 1, where ''k'' is an ''integer''. Zero's status as a natural number may be the subject of debate, but its status as an integer is not.—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 15:40, 3 March 2010 (UTC)
::::An unnaturally odd number then ;-) [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 18:52, 3 March 2010 (UTC)

== Density of primes ==

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. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:57, 20 March 2010 (UTC)
: It does mention it in the first paragraph, added a month ago: [http://en.wikipedia.org/enwiki/w/index.php?title=Prime_number&diff=345657481&oldid=345434764]--<small>[[User:JohnBlackburne|JohnBlackburne]]</small><sup>[[User_talk:JohnBlackburne|words]]</sup><sub style="margin-left:-2.0ex;">[[Special:Contributions/JohnBlackburne|deeds]]</sub> 22:03, 20 March 2010 (UTC)

:The lead already says: "the [[natural density|density]] of prime numbers within natural numbers is 0". The main result is the [[prime number theorem]] which is mentioned many times. Density 0 is a trivial consequence but a far weaker result. It probably has an easier proof but the main article for 100+ prime number articles is not the place for that. If there is a simple published proof from before PNT was proven then maybe it could be in [[Prime-counting function#History]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 22:11, 20 March 2010 (UTC)

It certainly has much easier proofs than that of the prime number theorem.  Some are very short and elementary&mdash;just a paragraph. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 01:24, 21 March 2010 (UTC)
== New graph ==

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 ''n''th prime <math>y=p_n</math> against the difference between squares of consecutive primes <math>x=p_{n+1}^2-p_n^2</math>. Since <math>p_{n+1}=p_n+2k</math>, you would expect the points on the grpah to lie along the lines <math>4ky=x-4k^2</math> 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. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 09:14, 28 March 2010 (UTC)

:I agree and have removed it. It only took the French Wikipedia 4 minutes to revert it.[http://fr.wikipedia.org/enwiki/w/index.php?title=Nombre_premier&diff=next&oldid=51616253] It was also added to the Russian Wikipedia. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 10:28, 28 March 2010 (UTC)

== n-th prime number formula? ==

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). [[User:Albmont|Albmont]] ([[User talk:Albmont|talk]]) 19:49, 14 April 2010 (UTC)

:See [[Prime number theorem#Approximations for the nth prime number]]. --[[User:Zundark|Zundark]] ([[User talk:Zundark|talk]]) 20:05, 14 April 2010 (UTC)

:For exact fomulae that are ridiculously slow to evaluate and amount to testing each number for primality, see [[Formula for primes]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 21:33, 16 April 2010 (UTC)
== Beginning of the universe ==

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?? [[User:Georgia guy|Georgia guy]] ([[User talk:Georgia guy|talk]]) 15:07, 4 May 2010 (UTC)

:There's a typo like that on page 24 of [<nowiki>http://www.amazon.com/Before-Beginning-Universe-Others-Helix/dp/0738200336</nowiki>] that seems to have escaped into the wild. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 15:33, 4 May 2010 (UTC)

::Any web site addresses in mind here? Are they [[WP:V|reliable sources]], because [http://eereporter.blogspot.com/2007/12/prime-numbers-driving-big-bang.html this one] references the idea and is weird and wonderful stuff.--'''''[[User:ianmacm|<span style="background:#88b;color:#cff;font-variant:small-caps">♦Ian<span style="background:#99c">Ma<span style="background:#aad">c</span></span>M♦</span>]] <sup>[[User_talk:ianmacm|(talk to me)]]</sup>''''' 15:36, 4 May 2010 (UTC)

:::I think the most relevant article on Wikipedia would be [[Deinstitutionalisation]] [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 22:22, 4 May 2010 (UTC)

== trial division ==

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  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Babahadi|Babahadi]] ([[User talk:Babahadi|talk]] • [[Special:Contributions/Babahadi|contribs]]) 13:39, 2 June 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:That's just a refinement of the "trial division" concept; if ''N'' can be written as ''a''<sup>2</sup>&nbsp;+&nbsp;''b''<sup>2</sup>, where ''a'' and ''b'' are relatively prime, then we need only look at primes of the form 4''n''&nbsp;+&nbsp;1, and so on.  — [[User:Arthur Rubin|Arthur Rubin]]  [[User talk:Arthur Rubin|(talk)]] 14:35, 2 June 2010 (UTC)

::Indeed. And the article does not say the simple trial division is <u>necessary</u> - it just says it is the simplest primality test. Also, if you want to improve the method, how do you know which integers less than sqrt(''n'') are prime ? Well, you can test each of them in turn by trial division - and then you have a recursive trial division algorithm. Or you eliminate all multiples of each prime that you find - then you have the [[ Sieve of Eratosthenes]]. Either way, you have gone beyond simple [[trial division]]. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 14:48, 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? [[Special:Contributions/94.74.158.16|94.74.158.16]] ([[User talk:94.74.158.16|talk]]) 03:31, 3 June 2010 (UTC)

:Yes, but this level of detail does not belong in the lead, which is supposed to be a short introduction to the article. There is a section called "Verifying primality" further down in the article that goes into more detail about various different primality testing algorithms. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 08:13, 3 June 2010 (UTC)
== Successive composite numbers ==

[[User:Ranjitr303|Ranjitr303]] ([[User talk: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 ?

:See [[Prime number#Gaps between primes]]. For any positive integer ''m'' &ge; 2, none of the ''m'' &minus; 1 numbers from ''m''! + 2 to ''m''! + ''m'' can be prime, because ''m''! + 2 is divisible by 2, ''m''! + 3 is divisible by 3, and so on. So to find a sequence of ''n'' consecutive composite numbers, just take ''m'' &ge; ''n'' + 1. For example, the 10 numbers from 11! + 2 = 39916802 to 11! + 10 = 39916811 are all composite numbers. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 08:03, 24 June 2010 (UTC)

== Irrationality of π implies infinite number of primes? ==

It is claimed that there is an equation

:<math> \frac{\pi}{4}\ =  (\frac{1}{1+\frac{1}{3}})( \frac{1}{1-\frac{1}{5}})( \frac{1}{1+\frac{1}{7}})(\frac{1}{1+\frac{1}{11}}) \cdots \; </math> 
(- 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:
:<math>\zeta(2) = \frac{\pi^2}{6}=\prod_{p \text{ prime}} \frac{1}{1-p^{-2}}  </math>
provides a much simpler proof from the transcendence of π to the infinitude of primes.  — [[User:Arthur Rubin|Arthur Rubin]]  [[User talk:Arthur Rubin|(talk)]] 18:21, 14 September 2010 (UTC)
:If we have already derived the Euler product for ζ, then there are much simpler ways of inferring infinity of primes than using the (fairly nontrivial) transcendence of π.—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 18:57, 14 September 2010 (UTC)
::I ''think'' the given equation has been derived by applying the Euler product to Gregory's series
:::<math>\frac{\pi}{4} = \sum_{k=0}^{\infty}\frac{1}{4k+1}-\sum_{k=0}^{\infty}\frac{1}{4k+3}</math>
::noting that each odd integer is the product of odd primes, and an integer of the form 4''k''+3 must have an odd number of prime factors of the form 4''k''+3 (counting repeated factors with appropriate multiplicity). But I agree that this claim definitely needs a reliable source, and if we are using Euler products then the non-convergence of the [[Harmonic series (mathematics)|harmonic series]] is a much more direct demonstration of the infinitude of primes. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 09:19, 15 September 2010 (UTC)

:If anything like this is to be in I'd have thought the best example would be the proof from that log ''x'' has no bound as ''x'' goes to infinity that is in 'Proofs from the Book'. I think one could make a case that proofs in that are notable. A proof from that pi is not rational is just silly considering how much work one needs to prove the antecedent. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 12:40, 15 September 2010 (UTC)

== By definition? ==

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.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/208.251.83.66|208.251.83.66]] ([[User talk:208.251.83.66|talk]]) 23:43, 27 September 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:Making 1 prime breaks uniqueness of prime factorization. See [[Prime number#Primality of one]].—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 10:26, 28 September 2010 (UTC)
== Generalized primes ==

[http://en.wikipedia.org/enwiki/w/index.php?title=Prime_number&action=historysubmit&diff=406146646&oldid=406069440 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. [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 23:31, 5 January 2011 (UTC)
:Note that this is being discussed at [[WT:WPM#Hoax warning]] and [[WT:WPM#International Journal of Algebra]].—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 11:06, 6 January 2011 (UTC)

== Euclid's theorem ==

I reverted {{ul|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. [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 21:59, 27 January 2011 (UTC)
:We're ''all'' here to make a better article but yes, I am sorry, it seems that I've been a bit hasty in reverting -I saw the Green-Tao discussion removed in the diff and the wording looked convoluted to me, so I reverted. I've seen the overall diff now between my revert and the current revision and I think it's much improved. My apologies! --[[User:Cyclopia|<span style="color:green;">Cycl</span><big>o</big><span style="color:green;">pia</span>]][[User talk:Cyclopia|<span style="color:red;"><sup>talk</sup></span>]] 22:45, 27 January 2011 (UTC)
::No prob! [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 22:46, 27 January 2011 (UTC)
== applications of primes? ==

As for applications of prime numbers, [http://mathoverflow.net/questions/985/applications-and-natural-occurrences-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?). [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 22:45, 31 January 2011 (UTC)

== possible further to do items ==

[[regular prime]], [[RSA number]], [[supersingular prime]], [[Adleman–Pomerance–Rumely primality test]], [[Giuga's conjecture]], [[Pepin's test]], [[Prime k-tuple]], [http://mathworld.wolfram.com/k-TupleConjecture.html 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]]

[[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 23:06, 31 January 2011 (UTC)
== Math formatting ==

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:
# Wikitext, with italic letters for variables (e.g. <nowiki>{{radic|''n''&amp;nbsp;&amp;minus;&amp;nbsp;1}}</nowiki> produces {{radic|''n''&nbsp;&minus;&nbsp;1}}). {{tl|nowrap}} or &amp;nbsp; should be used to prevent line breaks within formulas, and &amp;minus; rather than a hyphen should be used for subtraction.
# The {{tl|math}} template, with wikitext inside it (e.g. <nowiki>{{math|{{radic|''n'' &amp;minus; 1}}}}</nowiki> produces {{math|{{radic|''n'' &minus; 1}}}}). This automatically does the same thing as {{tl|nowrap}}, but it also uses a serif font instead of Wikipedia's default sans-serif. (As a minor technicality, the argument to the template needs to be preceded by "1=" if it contains an equal sign.)
# LaTeXed bitmap images of formulas, e.g. <nowiki><math>\sqrt{n-1}</math></nowiki> produces <math>\sqrt{n-1}</math>.
My personal preference is the {{tl|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 {{math|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 {{tl|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? —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:10, 28 January 2011 (UTC)
: I agree with David, the serif fonts are better for math.  Journal articles and textbooks do the same -- inline formulas use a slightly different font. <span style="white-space:nowrap;"><b>[[User:Justin W Smith|Justin W Smith]]</b> <i><sup>[[User talk:Justin W Smith|talk]]</sup>/<sub>[[Special:Contributions/Justin W Smith|stalk]]</sub></i></span> 00:28, 28 January 2011 (UTC)
:: I dislike the {{t1|math}} template, here and elsewhere.  Using <nowiki><math></nowiki> lets user preferences determine what they want to see: HTML, PNG, MathML, or a combination. [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 00:46, 28 January 2011 (UTC)
::: Most browsers do not show MathML so readers are stuck with an ugly PNG file stuck into the middle of a paragraph.  (This argument has been gone over so many times!) <span style="white-space:nowrap;"><b>[[User:Justin W Smith|Justin W Smith]]</b> <i><sup>[[User talk:Justin W Smith|talk]]</sup>/<sub>[[Special:Contributions/Justin W Smith|stalk]]</sub></i></span> 00:52, 28 January 2011 (UTC)
::: BTW, I'm using Google's Chrome browser (which supposedly supports MathML?) and I have my Wikipedia preferences set to use MathML, but <math>\sqrt{n-1}</math> still shows up as an image. <span style="white-space:nowrap;"><b>[[User:Justin W Smith|Justin W Smith]]</b> <i><sup>[[User talk:Justin W Smith|talk]]</sup>/<sub>[[Special:Contributions/Justin W Smith|stalk]]</sub></i></span> 00:57, 28 January 2011 (UTC)
:::: I'm also using Chrome and I have my preferences set to force png rendering even when Wikipedia thinks it should render in HTML instead.  So preferences clearly vary -- your ugly is my strong preference. [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 01:10, 28 January 2011 (UTC)
::::: The way I see it &mdash; If the blocky PNG-style rendering were generally considered "better", then you would see that style of rendering used in journals and textbooks. <span style="white-space:nowrap;"><b>[[User:Justin W Smith|Justin W Smith]]</b> <i><sup>[[User talk:Justin W Smith|talk]]</sup>/<sub>[[Special:Contributions/Justin W Smith|stalk]]</sub></i></span> 01:15, 28 January 2011 (UTC)
:::::: What you see in journals is [La]Tex which is emulated by Wikipedia's <nowiki><math></nowiki> tag -- is this blahTeX?  You certainly don't see MathML-based rendering in journals.  So in essence, this is a false choice: we don't have the option of using what publishers use, and the closest we come to it is with the blahTeX.  Of course I don't see that as an argument for its use (or disuse), but since you brought it up... [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 01:48, 28 January 2011 (UTC)
:::::::Well the radical with the text version is horrible so I'd go with the Latex formatting. I just wish MathML was working well rather than being experimental. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 07:46, 28 January 2011 (UTC)
::::::::I wish we had [[MathJax]]. But we don't. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 07:50, 28 January 2011 (UTC)
:::: I have verified the bug Justin mentioned and noticed that [https://bugzilla.wikimedia.org/show_bug.cgi?id=25646 this bug has been reported]. —[[User:Kennyluck|Kennyluck]] ([[User talk:Kennyluck|talk]]) 23:08, 9 February 2011 (UTC)

:(unindent) I personally prefer HTML markup whereever this is possible, just because it blends in most smoothly. (In the case above with the square root, the horizontal overline does not quite connect to the one of the radix, so in this particular case I would use TeX probably.) The point that l and I are indistinguishable in HTML is a fair point, but actually I never encountered this problem before; even if so, it is easily possible to choose letters which avoid this particular ambiguity between l and I. [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 09:55, 28 January 2011 (UTC)
:: When <nowiki><math></nowiki> is working properly it will render simple formulas in HTML unless you have it set to disallow that.  Also it should transition into MathJax nicely when/if Wikipedia moves to it.  Of course this supposes that no hacks like \, are used to 'force' png rendering. [[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 19:33, 28 January 2011 (UTC)

== Euclid's proof of the infinitude of primes ==

[http://en.wikipedia.org/enwiki/w/index.php?title=Prime_number&diff=next&oldid=410446573 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. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 02:46, 29 January 2011 (UTC)
: Also, I would like to know why it is advantages to rephrase Euclid's proof into modern notation when the section purports to be about what Euclid did.  Euclid did not use that notation. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 02:56, 29 January 2011 (UTC)
::Thanks for fixing it! My bad... I'm really not good at history. As for the notation: in order to make it as accessible as possible, I prefer modern notation, but a note about Euclid's original notation might be worth it. [[User:Jakob.scholbach|Jakob.scholbach]] ([[User talk:Jakob.scholbach|talk]]) 16:59, 31 January 2011 (UTC)

::The joint paper by Hardy/Woodgold in Mathematical Intelligencer demolishes no myth of Euclid's proof for the simple reason that the authors never displayed a valid Direct versus Indirect to compare and to thence show that Euclid did one and not the other. All that Hardy/Woodgold showed was a few pieces of opinion and plea-for-evidence that Euclid probably did a Direct method, but not a "deciding evidence". In order for a deciding evidence, one has to be able to do a valid Euclid Infinitude of Primes, indirect method, and Hardy/Woodgold never were able to show valid indirect method. In the valid Indirect method, Euclid's number is necessarily a new prime and that is the deciding piece of evidence because in Euclid's ancient proof, Euclid searches for a prime factor, which means, positively and without doubt that Euclid did a Direct method. So that Hardy/Woodgold/Davis of Mathematical Intelligencer have only provided a "opinion" that Euclid did a Direct method and not given us deciding-evidence. Until the day that Hardy/Woodgold/Davis actually produce their own valid indirect method of Euclid Infinitude of Primes proof, they are out of bounds on their claims of Euclid. [[Special:Contributions/216.16.55.212|216.16.55.212]] ([[User talk:216.16.55.212|talk]]) 20:38, 10 February 2011 (UTC)Archimedes Plutonium

::Also, it is evident that the Euclid proof as seen on the cover page has a flaw of not having the definition of prime as its beginning step, which makes the proof invalid since it is the anchor of the proof and that the claim Euclid's number is possibly prime is not justified. It is omissions like this that indicate that Hardy/Woodgold/Davis lack the abilities to be the spokespersons for the Euclid proof. And another big mistake is the discussion by M. Hardy says "Although the proof as a whole is not by contradiction, in that it  does not begin by assuming that only finitely many primes exist, there  is a proof by contradiction within it: that is the proof that none of  the initially considered primes can divide the number called q above." This is another M. Hardy opinion, for a valid Direct has no such subclause contradiction. So what is a statistician of M. Hardy and a electrical engineer of Catherine Woodgold doing puttering around on a slice of mathematics that needs a Logician? A logician that can discuss in depth what the reductio ad absurdum method truly is, and not someone confused with contrapositive as compared to reductio ad absurdum, and a Logician that can freely and easily display his own Indirect method proof, and not like M. Hardy that never displays his own indirect.  [[Special:Contributions/216.16.55.212|216.16.55.212]] ([[User talk:216.16.55.212|talk]]) 20:38, 10 February 2011 (UTC)Archimedes Plutonium

:::I believe this really is Archimedes Plutonium and those proofs  which were removed demonstrated it was him. So who exactly would be claiming copyright on them? [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 11:31, 17 February 2011 (UTC)
:::The explanation seemingly is that we cannot know that a person who just signs themselves as someone here is the same as a person who signs themselves on a usergroup posting. It was rather a long clip. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 12:12, 17 February 2011 (UTC)