Talk:Prime number/Archive 8

semi-protected edit request
Add prime number 343 to list of primes. — Preceding unsigned comment added by 95.49.26.86 (talk) 11:57, 5 July 2014 (UTC)
 * Not prime: 343 = 73. Paul August &#9742; 14:25, 5 July 2014 (UTC)

It is a mystery why anyone would turn up on this talk page and say that low value prime numbers are missing, because all of the primes in this range were discovered a long time ago. It is possible, however, that there are non-Mersenne primes in between the very large ones that have been discovered by computer.-- ♦Ian Ma c M♦  (talk to me) 19:52, 7 July 2014 (UTC)
 * For an even stranger example, see Grothendieck prime. —David Eppstein (talk) 20:50, 7 July 2014 (UTC)
 * Maybe it was Alexander Grothendieck who posted the Semi-protected edit request on 14 May 2014 (see above). I've often wondered how long it will take to have a guaranteed complete list of all of the prime numbers in the currently known range. The current list is not guaranteed to be complete, either for Mersenne primes or ordinary primes.-- ♦Ian Ma c M♦  (talk to me) 21:24, 7 July 2014 (UTC)
 * Do you mean the adding composite 57 to the list 164.126.206.226 (talk) 13:40, 10 July 2014 (UTC)?

Rational Primes
Is there a generally agreed on definition of a rational prime number? 173.79.197.184 (talk) 13:08, 1 August 2014 (UTC)
 * If you mean a rational number that cannot be factored into smaller rational numbers, then, no. There is no meaningful definition for such a thing, since every rational number is a unit.  Since every rational number has a multiplicative inverse, any factorization could be modified to include smaller rational numbers and their inverses, ad infinitum. — Anita5192 (talk) 17:29, 1 August 2014 (UTC)
 * "Rational prime" can be used refer to the usual concept of prime number in distinction to irreducible elements or prime ideals of rings of algebraic integers. See for example   Deltahedron (talk) 18:14, 1 August 2014 (UTC)
 * It is also the case that every positive rational number can be uniquely expressed as the product of (the usual) primes, just with the possibility of negative exponents as well as positive. Would the article be improved by the mention of either of these points?  Deltahedron (talk) 18:24, 1 August 2014 (UTC)
 * This is an interesting extension of the concept of unique factorization of natural numbers, and it would not harm to mention it. It is most closely tied in with the Fundamental theorem of arithmetic, perhaps present it as an incidental associated uniqueness factoring?  It should be made clear that the primes are not primes in Q, though. —Quondum 20:51, 1 August 2014 (UTC)

Miller-Rabin
Third to last paragraph in History should not ignore Miller-Rabin: http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html. Still the gold-standard test for speed and practicality. 19:33, 7 August 2014 (UTC)


 * That paragraph is about primality proving. Miller-Rabin is one of many probable prime tests, and it isn't old. I don't think it belongs in a brief history section going back millenia. Fermat's little theorem is historically much more important and already mentioned earlier. PrimeHunter (talk) 21:44, 7 August 2014 (UTC)

The most basic sieve: 6n±1
This really is the most basic element of prime distribution. I don't understand why anyone would want to hide this very basic and important fact about primes. As I said: "The 6n±1 is the most primary filter, and is basic to all elementary introductions to primes. This 2/3 filter does NOT exist in any other mod that isn't a multiple of six. If you want to describe the 6n±1 concept in another way, go for it." This is probably more important and basic than most everything else on this page. Why would you want to delete it? Again, if you don't like the way I've illustrated the 6n±1 concept, feel free to improve upon it. But deleting it is just plain weird... TurilCronburg (talk) 20:48, 19 July 2014 (UTC)
 * welcome! Can you please provide a source that expresses the opinion that this particular sieve is unique? That would be helpful, and might convince other editors that this merits inclusion. Also, please review our policy on edit warring. VQuakr (talk) 21:04, 19 July 2014 (UTC)

Wolfram includes it in their Prime Number write-up. 67.188.92.176 (talk) 20:05, 10 August 2014 (UTC) Robin Randall, Aug 2015

This sort of thing is literally as basic as it gets. It's the kind of thing you'd see on an elementary school level discussion of primes. For example: http://primes.utm.edu/notes/faq/six.html which is the prime FAQ on a kid's math website. This fact about primes is kind of the equivalent of saying that atoms are made up of protons, neutrons, and electrons.TurilCronburg (talk) 21:26, 19 July 2014 (UTC)


 * I'll be blunt. It is just one of an infinite sequence of such filters, and it is not the "most basic" or first such filter. A more basic filters would be ≠2n and ≠3n. The ≠5n filters is next. 6n±1 filter is a composite of the ≠2n and ≠3n filters. Any discussion of filtering would have to deal with the whole topic under the correct heading; this is already covered under §Sieves. Further, filtering and distribution are distinct concepts, and should not be confused. Please also read WP:NOR. —Quondum 21:28, 19 July 2014 (UTC)


 * It is mentioned as a method of compressing the tabulation of primes in  Deltahedron (talk) 21:39, 19 July 2014 (UTC)


 * I'll be even blunter Quondum, the 6n±1 fact is probably the most interesting, understandable, and useful bit of information on the whole page after the introduction. ALL primes after 3 are within one of a multiple of 6.  Other than primes all being odd (other than 2), this is the only simple pattern that exists in the distribution of primes.  It doesn't work with mod 5.  Or mod 4.  Or mod 7.  Or any other modulo that isn't a multiple of 6 (or 2, but that's pretty uninteresting).  And the distribution of the primes is entirely relevant to the category of distribution.  And my edit follows the lovely quote about primes being both random and predictable, so it's clearly in a good spot.  Removing it is inexplicable, as far as I'm seeing.TurilCronburg (talk) 19:31, 25 July 2014 (UTC)


 * Continuing the bluntness theme — that's your personal opinion. When you find your opinion opposed to the consenus of other editors, then it's time to yield gracefully.  Deltahedron (talk) 19:38, 25 July 2014 (UTC)


 * On the contrary, it works as a modulo of any number N that's a product of two or more unique primes, in that any prime number mod N cannot be a multiple of any of the primes that multiply to N. In the case of 6 with primes 2 and 3, it eliminates 0, 2, 3, and 4, leaving only 1 and 5, but you could do the same with 10, primes 2 and 5, leaving only 1, 3, 7, and 9, or with 15, primes 3 and 5, leaving 1, 2, 4, 7, 8, 11, 13 and 14.  The fact that these are readily produced from any starting set of prime numbers makes them unnotable enough to be not worth mentioning.  The primorials might be marginally more notable in that regard but it's nothing I'd push for if the consensus doesn't support it.  mwalimu59 (talk) 21:04, 25 July 2014 (UTC)


 * Yes, this is my opinion, as a teacher. This most simple fact, which I've cited a resource for (as requested) is very important and useful to include in a basic, encyclopedia form introduction to the prime numbers and their distribution.  If you don't agree, that's fine, but clearly others do find this fact very interesting, so why delete it?  Also, can you name any other modulo other than 2 (not that interesting :-) and 6 where a single number defines the location of nearly all of the primes?  The example of mod 10 gives us primes all over the place, not around a single number.  Same with all the other modulos.  Again, while it might not be interesting to you, it is to many students just starting to explore primes.  So including this fact is valuable and improves the article.TurilCronburg (talk) 21:25, 25 July 2014 (UTC)


 * I can name a modulus other than 6 which works just as well as 6. All primes greater than 2 are of the form 4n ± 1.  That this is true is trivial to prove.  Proving that there are an infinite number of primes of both forms, i.e., 4n + 1 and 4n - 1, is a sequence of exercises in my number theory textbook.  So, you see, 6 as a modulus is not unique.—Anita5192 (talk) 06:43, 26 July 2014 (UTC)


 * Anita, mod 4 only eliminates 1/2 of non-primes, so it doesn't work as well as mod 6. Also, mod 4 is essentially exactly the same as mod 2, which isn't very interesting.  (But that fact IS included in the entry, and I presume you aren't going to delete that fact, right?)  You don't see people constantly asking about whether or not all primes (after a certain low number) are odd, while you do see the question about primes being within one of a multiple of six regularly.  So this fact is clearly important to many people, as noted on the link I provided as the citation, from Drexler University's math website, and it's only logical that the fact is an important one to include on a general introduction to the topic, which an encyclopedia entry is supposed to be. TurilCronburg (talk) 16:10, 26 July 2014 (UTC)


 * It would help TC make the case for the importance of this material to find independent reliable sources that mention it and which in turn explain its significance: I gave one example above. Simply reiterating a personal opinion on its importance does not help.  Deltahedron (talk) 07:12, 26 July 2014 (UTC)


 * Deltahedron, I did include a reference in my last update, as you suggested, from Drexler University's math website. But you deleted my contribution again.  So clearly there is some other reason for you censoring this basic fact.  I really don't understand what's going on here... TurilCronburg (talk) 16:10, 26 July 2014 (UTC)


 * Nobody is "censoring" this fact. The community of editors interested in this article would like to have the opportunity to discuss the material.  Your view that it is important may well prevail if you make the case clearly and don't try to short-circuit the consensus-building process.  Using emotive language like "censoring" does not help you make your case.  Deltahedron (talk) 16:16, 26 July 2014 (UTC)

An observation, particularly directed at TurilCronburg: one reason that your contribution is meeting resistance is that it is misplaced. When mathematicians speak of "the distribution of primes", they are not generally speaking about modular identities (like, e.g., the fact that all primes other than 2 are odd). Instead, the phrase "distribution of primes" relates to questions like "what is the probability that a randomly selected 10-digit integer is prime?" Reasonable answers to this question are very hard, and require sieving by larger and larger sets of primes; meanwhile, for any finite collection of primes it is easy to write down a modular sieve of exactly the sort you are describing. I agree with you that the mod 6 sieve is more appealing than a random example of such a thing; it is totally plausible that one could find some interesting history on the use of this sieve and slot a paragraph about it into this article or a related one. But, the place you are putting this paragraph is definitely wrong.

It may help you to imagine what would happen in the future were some other editor to come and insist that we include mention of the important and fascinating fact that every prime number is congruent to plus-or-minus 1,2,4,7,8,...,37 mod 75. How should someone explain to this editor (who has a deep and abiding personal belief in the importance of this fact) why it does not belong in the section on distribution of prime numbers? Best, JBL (talk) 18:47, 26 July 2014 (UTC)

Prime Numbers in Nature
For Quantum connection to zeta function could it be referring to the Casimir Effect? 67.188.92.176 (talk) 20:05, 10 August 2014 (UTC)Robin Randall, Aug. 2014

"no known useful formula"
"There is no known useful formula that sets apart all of the prime numbers" - is there a useless one? What does "useful" mean? Are there formulas that require computing power beyond our abilities? --Richardson mcphillips (talk) 17:52, 6 February 2015 (UTC)
 * There are formulas that are not useful because they involve functions that are defined in terms of prime numbers, so using them to compute prime numbers would be circular reasoning. —David Eppstein (talk) 18:56, 6 February 2015 (UTC)
 * There are also formulas whose application requires so much computation that it would be faster to use trial division. For example, Wilson's theorem. —Mark Dominus (talk) 20:53, 6 February 2015 (UTC)
 * The lead is only supposed to summarize the article. Prime number goes into more detail. It's hard to give a short precise statement (or even a long one) about what is lacking to have a "good" formula but such a lack is often mentioned, and lots of people have searched and still search for it, whatever "it" is. I think it's appropriate to give some hint in the lead that we don't have the kind of formula we would like. PrimeHunter (talk) 01:47, 7 February 2015 (UTC)

Prevalence of primes at a given level
Under "number of prime numbers below a given number" section, you might want to add the derivative of pi(n) n/ln(n) which is [ln(n)-1]/71.139.161.9 (talk) 19:37, 7 September 2014 (UTC)
 * This might be true, but it is something of a statement of the obvious. — Preceding unsigned comment added by 86.181.91.77 (talk) 13:49, 15 January 2015 (UTC)
 * It is also unnecessarily complicated, as the simpler 1/ln(n) is asymptotically the same, and it's only an asymptotic formula anyway. The editor who uses the pseudonym "JamesBWatson" (talk) 15:13, 24 February 2015 (UTC)

New section: The Interval Containing At Least One Prime Number needed.
It seems like someone forgot to have a section on the Interval Containing At Least One Prime Number. So, I will suggest it. John W. Nicholson (talk) 17:55, 1 March 2015 (UTC)


 * As most people are not mind-readers, probably it would be a good idea to include in your suggestion enough information for other editors to have some clue what you are talking about. Something related to prime gaps?  --JBL (talk) 18:01, 1 March 2015 (UTC)


 * An example from Dusart 2010, for x>= 396738, the interval [x, x + x/(25ln^2(x))] contains at least one prime. I know there is a history of other statements like this. John W. Nicholson (talk) 19:38, 18 March 2015 (UTC)

Article contradicts with other wikipedia article.
The article claims that the latest prime number was found in April 2014 and later in the sentence links to the wikipedia list of largest prime numbers. However, that article claims that the latest largest prime number was discovered in February 2013. 216.96.200.76 (talk) 16:13, 14 April 2015 (UTC)


 * The article currently says "As of April 2014, the largest known prime number has 17,425,170 decimal digits." Here "As of" isn't meant to be a discovery date but a date where the statement was known to be valid (it's also valid today). If it only said "The largest known prime number has 17,425,170 decimal digits", then it would become invalid when the record is broken. The record would probably quickly be updated in our own article but we also have many reusers who copy our articles without updating their copies. See more at As of. PrimeHunter (talk) 16:31, 14 April 2015 (UTC)

Maybe I'm misunderstanding but ...
" ...the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n."

To me this seems to be saying that any two digit number has a 50% chance of being prime, a three digit number has a 1 in 3 chance, etc. Obviously, that's not the case.

Am I just misreading it? — Preceding unsigned comment added by Tym King (talk • contribs)


 * There's a constant of proportionality you're missing, so the probabilities are somewhat smaller than what you state. And also, "probability" is a weird thing to talk about for a deterministic concept like whether a specfic number iis prime — it's more like, if you chose a k-digit number at random, the probability that your random choiice happens to be prime is approximately c/k where c is that constant of proportionality. —David Eppstein (talk) 14:54, 18 June 2015 (UTC)


 * Here c = 1/log(10) = 0.43429... PrimeHunter (talk) 20:01, 18 June 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Prime number. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive https://web.archive.org/20071020141624/http://maa.org:80/mathland/mathtrek_6_28_99.html to http://www.maa.org/mathland/mathtrek_6_28_99.html

When you have finished reviewing my changes, please set the checked parameter below to true to let others know.

Cheers. —cyberbot II  Talk to my owner :Online 10:06, 27 August 2015 (UTC)

First Paragraph
I believe my addition of : "non-prime numbers (or rather, their factorization) can be said to be 'composed' of the primes themselves" has merit, as it makes clear the explanation that immediately follows it. Can we not leave it in? — Preceding unsigned comment added by Baguettes (talk • contribs) 21:57, 26 September 2015 (UTC)
 * I oppose it. The placement in the second sentence of an article about prime numbers (and not composite number) is too prominent, and it only makes sense together with an explanation that comes later and already works fine alone. Do you have a source saying the word "composite" was chosen with prime factorization in mind and not factorizations in general like for example 60 = 6×10? PrimeHunter (talk) 22:49, 26 September 2015 (UTC)
 * I've done a search for sources and I kinda see what you mean. Though the sources immediately to hand indicate that Composite Numbers can be 'decomposed', that decomposition isn't necessarily into Primes (as my sentence eluded to), but rather into at least one pair of other natural numbers, which may themselves be composite.  Surely, my sentence could be rewritten to account for this, though? Baguettes (talk) 19:01, 27 September 2015 (UTC)
 * It could be rewritten but I would still oppose it in the first paragraph, and it has already been removed by three different editors who didn't comment on this particular issue. PrimeHunter (talk) 11:14, 28 September 2015 (UTC)

Less than or equal
Again, I am confused when it says pi(11) = 5 for the 5 primes LESS THAN OR EQUAL to 11. Later it appears pi(n) represents the number of primes LESS THAN the number N. Which is it? Or does the approximate operator take care of this?RobinLRandall 15:01, 15 Oct 2015 (UTC) — Preceding unsigned comment added by 67.188.92.176 (talk)


 * pi(x) is the number of primes less than or equal to x. For approximations like x/log(x), it doesn't matter whether it's called an approximation to pi(x) or to the number of primes less than x. The two counts will either be identical or one apart, and it's just an approximation. If you don't use the named function pi(x) then "less than x" is easier to say than "less than or equal to x". That may be a practical reason for omitting the unnecessary "or equal to". PrimeHunter (talk) 22:48, 15 October 2015 (UTC)

References to Weisstein's "mathworld" should be avoided
I just removed a reference to Mathworld regarding the fact that Euler did not consider 1 as a prime number, and replaced it by a direct reference to one of Euler's theorems, from which statement this fact is clear. In fact Weisstein's reference only mentioned that Goldbach considered 1 to be a prime, and not that Euler did not. But this is beside the point. I think we should generally avoid references to Weisstein's online encyclopedy, which is not reliable at all as it contains many approximate and false informations. It has the same disadvantages as Wikipedia has (of having subjects treated by non specialists), with the further drawback of being edited by essentially one single non specialist - Weisstein himself. Sapphorain (talk) 08:02, 18 November 2015 (UTC)
 * I agree. MathWorld can be ok as an external link but best avoided as a reference. Also, thanks for tracking down this reference — the part about Goldbach treating 1 as prime is easy to source but Euler not seemed more difficult to me. —David Eppstein (talk) 08:10, 18 November 2015 (UTC)

Recent edits to lead
There has been some recent edit/reversion going on. My feeling is that it would be nice to mention the AKS algorithm (or some other aspect of algorithmics) in the paragraph in question, but that the level of technicality should be kept to the level "could be understood with a high school education." I suggest discussing the text here in order to reach consensus, rather than continuing to edit the article directly. --JBL (talk) 22:21, 5 November 2015 (UTC)
 * I don't think that the last edit was too technical. I would say too complicated. I am including it here to aid improvement efforts:

"The AKS algorithm, which is considered very slow for practical sizes, shows that deciding whether a number $n$ is prime has polynomial complexity, which means that in the general case it can be done quickly. For practical sizes, probabilistic algorithms such as the Miller-Rabin algorithm are preferred, which execute very fast at the cost of some small error probability."
 * Nxavar (talk) 10:35, 6 November 2015 (UTC)
 * "can be done quickly" is vague, and nobody uses AKS in practice even if they want primality proofs. Other methods like ECPP are far faster but haven't been proved to have polynomial complexity. ECPP has proved a 30,950-digit prime. That size might take billions of times longer with AKS if any implementation can handle it. PrimeHunter (talk) 00:43, 7 November 2015 (UTC)
 * ECPP is a great algorithm. However, it is not know whether it is in P. This is why "quickly in the general case" is correct for AKS and unknown for ECPP. Also, the 30,950-digit prime is a special prime, not a general prime, that is there might be a prime with that many digits that takes more time to test with ECPP than it does with AKS. Considerations like this is what makes AKS such an important finding Nxavar (talk) 09:06, 9 November 2015 (UTC)


 * According to the publication "", it is fine to call AKS a fast algorithm from a mathematical perspective. The wikipedia article on prime numbers is an encyclopedic article and what the subject specialist think should be taken seriously. In my edit I make it clear that the AKS has limited practical value and given the above publication I believe there is enough ground to include the edit in the lead. Nxavar (talk) 16:01, 10 December 2015 (UTC)

Surprisingly
Hi, this is a ridiculously minor thing, but I think the word "surprisingly" should be deleted from the bit about Ulam spirals clustering on certain diagonals, so as to maintain an objective tone. — Preceding unsigned comment added by 140.233.173.75 (talk) 10:00, 18 December 2015 (UTC)


 * I am not certain whether the word made much difference, but I removed it anyway and cleaned up the other wording in that section. — Anita5192 (talk) 17:08, 18 December 2015 (UTC)


 * I agree the word shouldn't be there. If the word was intended to say that there is no theory to explain that behavior, then that should be stated more explicitly, with a source. Gap9551 (talk) 17:29, 18 December 2015 (UTC)
 * Right, and there is nothing surprising to mathematicians who understand the effect of avoiding small prime factors like some diagonals do. Some sources describe it with hype but so far it behaves as expected, although it's unproven whether it will continue to behave like that when n tends to infinite. PrimeHunter (talk) 22:09, 18 December 2015 (UTC)

Incorrect statement
I'm just a random passer-by, but want to point out that the statement below on the page,  about the prime number theorem,  is not correct:

..., which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.

The prime number theorem defines \pi(x)   as  the number of primes less than x,  and says that is \pi(x) is  proportional to log(x).

It says that the probability that a number n  chosen at random from [2 ... x] is prime,  is proportional to y = log(x).

This is not the same thing as the number of digits of  n, the number that was chosen at random. The probability that a random number of exactly n-digits is prime,  is proportional to something like  \pi(10^n) - \pi(10^n-1). You have to cut out all the numbers that have fewer digits from the pool. — Preceding unsigned comment added by Ccmxxx (talk • contribs) 16:18, 26 December 2015 (UTC)
 * That correction to the details of the calculation doesn't change the inverse-logarithmic proportionality. —David Eppstein (talk) 18:42, 26 December 2015 (UTC)

Wording re regular polygon construction
My edit has been reverted, restoring the following wording:


 * A regular n-gon is constructible using straightedge and compass if and only if
 * n = 2i · m
 * where m is a product of any number of distinct Fermat primes and i is any natural number, including zero.

True, the m=1 case is just a null product. But the point of my edit was that the old and restored version claims that (m, n)=(1, 0) or (1, 1) gives the number of sides of a constructible regular polygon: a1-sided polygon and a 2-sided polygon respectively. The former is impossible, and the latter is possible only if we admit a degenerate polygon. So I think the wording needs to be corrected. Loraof (talk) 21:12, 15 March 2016 (UTC)


 * After David Eppstein dealt with the issue you raised, you have made a second change to make the phrasing more complicated in order to avoid the empty product -- why? (I also am not certain that your new phrasing is correct -- haven't you lost n=3?) Frankly I do not see what the problem with the original wording is -- even in the worst case (if I agree that there is no such thing as a regular  1-gon or 2-gon), we are making a true assertion about an empty set (that all its members have the property of constructability).
 * On an unrelated note, could someone move the paragraph and image about Fermat primes and constructability into the section on Applications below? It obviously does not belong where it is now.  (I am on Amtrak and the connection is not good enough for me to edit the article.) Otherwise I will try to get to it in the next few days. Thanks in advance. --JBL (talk) 02:32, 16 March 2016 (UTC)
 * My feeling is also that the 2^i m formula is entirely adequate to encompass all cases, using the empty product, and that adding cases to the characterization to avoid the empty product is a mistake (it makes the result harder to understand, not easier). —David Eppstein (talk) 03:27, 16 March 2016 (UTC)

Euler's proof
This article and Euclid's theorem attribute different proofs of the infinitude of primes to Euler. This one says it's about the divergence of the series of prime reciprocals, while the other one involves an Euler product formula for the harmonic series. Are they both Euler's? Shouldn't we be more consistent? —David Eppstein (talk) 21:48, 17 March 2016 (UTC)
 * I think there is only one proof, of the divergence of the series of prime reciprocals, which makes use of the product formula for the harmonic series: it is contained in "Variae observationes circa series infinites" . Euler establishes through a product formula that the harmonic series is the (hyperbolic) logarithm of infinity ("absolutus infinitus", which he considers as a somehow special number) in Theorema 7. Then he uses the latter to prove in Theorema 19 that the sum of the reciprocal of primes is (hyperbolic) log log of the infinity. Sapphorain (talk) 23:31, 17 March 2016 (UTC)
 * They are closely related but I'm not convinced that they are really the same proof. To go from divergence of harmonic series to divergence of reciprocal prime series, one uses the product formula, takes the log of the resultiing product, and then argues that the error terms in the log are inconsequential. But the proof at Euclid's theorem stops after using the product formula, arguing that the finitude of primes would already make the product finite, without bothering to take its log. So there's a disconnect between what we say at that article (which doesn't mention the reciprocal prime series) and what we say here (which doesn't mention the harmonic series). —David Eppstein (talk) 18:43, 18 March 2016 (UTC)
 * Yes, you are right. Euler already mentions as a consequence of his Theorem 7 the infinity of primes, in the Corollary 2 page 174 (in which he remarks they are more numerous than squares): "…sequitur infinities plures esse numeros primos, quam quadratos…" Thus after his Theorem 19 he has of course no reason to repeat that there is an infinity of primes. So the two references to Euler you mention are related to the same proof, but only the assertion that Euler obtained a new proof of Euclid's theorem after proving the product formula (Theorem 7) seems to be appropriate; the assertion that he obtained it after proving the infinity of the sum of reciprocal of primes (Theorem 19) appears to be irrelevant, and to have been popularized by people who (like me) didn't read Euler's proof in details. Sapphorain (talk) 00:07, 19 March 2016 (UTC)

Semi-protected edit request on 2 June 2016
Number of prime numbers

Prime Numbers Set Definition

Definition of prime numbers can be written all over as this: Prime numbers does not have a factor other than itself and 1; as following;


 * For $$N \in Z^+ $$;


 * $$ a_i $$ $$ i^{th} $$ row value, $$ b_j$$ $$j^{th}$$ column value which $$ i, j = 1, 2, 3, \dots N. $$


 * $$ a_i, b_j = 1, 2, 3, \dots N.$$


 * $$ x_{ij} = a_i * b_j $$

Sets consists of $$ x_{ij} $$'s are;


 * $$A = \{ x_{ij}; \quad i, j \in Z^+ \}$$
 * $$B = \{ x_{ij}; \quad i, j \in Z^+ - \{1\} \}$$

Thus;
 * $$A = \{u_1, u_2, u_3, \dots \}$$


 * $$B = \{v_1, v_2, v_3,\dots \}$$

P shows prime numbers' set,


 * $$P = A - B - \{1\}$$

Some features of sets are like below,


 * $$ s(A) > s(B). $$
 * $$ B \subset A. $$
 * $$ A = B \cup P \cup \{ 1 \}. $$

Set A consists of positive integers starting from 1, set B consists of positive integers greater than 1. Separately multiplying their own members with their own thus getting a new pair of sets. The difference of the new sets is the set of almost prime numbers. 1 is not a member of the prime numbers' set, prime numbers' set could be obtained by substracting 1 from latest found set. If it's desired a N value could be selected by that prime numbers' set can be obtained within the range of 1 to N --Nexusiot (talk) 20:46, 2 June 2016 (UTC) Comment : "Please add this new feature" --Nexusiot (talk) 20:46, 2 June 2016 (UTC)

Nexusiot (talk) 20:46, 2 June 2016 (UTC)
 * Red information icon with gradient background.svg Not done: For several reasons.


 * Your account will be  in about 3 days, at which time, you will be able to make the edit.
 * This request is unclear exactly what needs to be changed or added. It would be easier to process this if you wrote this in a "Change X to Y" format. — Andy W.  ( talk  · ctb) 21:34, 2 June 2016 (UTC)

Prime number formula
All prime numbers have to follow this formula is: 6n±1 — Preceding unsigned comment added by 212.253.111.210 (talk) 00:33, 5 October 2016 (UTC)

Best regards Nedim ERDAN — Preceding unsigned comment added by 212.253.111.210 (talk) 00:36, 5 October 2016 (UTC)


 * 2 and 3 don't. Numbers of form 6n±1 is the same as numbers not divisible by 2 or 3 so it clearly includes all primes above 3. PrimeHunter (talk) 00:52, 5 October 2016 (UTC)
 * This was discussed here in the talk page archive.-- ♦Ian Ma c M♦  (talk to me) 15:38, 19 October 2016 (UTC)

101
In the list of prime number you have 101 as a prime number, this is incorrect. — Preceding unsigned comment added by Theisencouple (talk • contribs)
 * 101 is certainly a prime. See A000040 or test it yourself. PrimeHunter (talk) 18:33, 19 November 2016 (UTC)

Largest known Proth prime
The largest known proth prime in Prime_number is not up-to-date, see also Seventeen or Bust. Due to the protection of the article I can't update it myself. Der Waldkauz (talk) 12:49, 3 January 2017 (UTC)
 * Thanks. I have updated this and other records.[//en.wikipedia.org/w/index.php?title=Prime_number&diff=758100141&oldid=752956371] PrimeHunter (talk) 13:53, 3 January 2017 (UTC)

Prime number (final ?) formula
I think will be time to give a math definition of primes that match as possible the "talking of" definition:

$$ n\in \mathbb{N^+}; n>2 ; n=odd ; $$ is a prime if and only if:

$$ z = n! / n^2 = (n-1)! / n \in \mathbb{(Q-N)} $$

This definition respect the known one in the main concept of Prime as result of a recursive division, missing just 2.

From $$n>4$$ the definition fits for all Naturals.

Usinig this definition and 2 algos involving Sums / Product / Fractions and Integer part of..., it's possible to count the primes and given one find the next one. This algos are of course not computable immediately from very little primes due to factorial product, but in theory they give a perfect definition of Primes as well sorted numbers. So pls delete elsewhere the phrase "we actually don't know if primes are well sorted or not...".

Stefano Maruelli — Preceding unsigned comment added by StefanoMaruelli (talk • contribs)
 * Wikipedia is based on reliable sources. We shouldn't invent a definition not used by any sources. It would also be an odd way to define primes. We are a general encyclopedia and don't need definitions to use pure mathematical notation many readers wouldn't understand. If we did want it then we would formulate the normal definition like that. PrimeHunter (talk) 14:43, 21 January 2017 (UTC)


 * The statement is true, not hard to prove, and certainly not as useful as you seem to think. It is no more a "formula" than the usual definition in terms of number of factors. --JBL (talk) 15:02, 21 January 2017 (UTC)
 * I get no search results on "we actually don't know if primes are well sorted or not...". If it actually says something else then please don't falsify quotes when you refer to text. PrimeHunter (talk) 16:33, 21 January 2017 (UTC)

Semi-protected edit request on 22 January 2017
Please add in the References section the information about my new book. Please add in the External links section the link-information about my new book (hope this is correct, because this is my first post ....) deeds 09:45, 22 January 2017 (UTC)
 * I should like to point the fact that neither the Mathematical Reviews of the AMS nor the Zentralblatt MATH reviewed or indexed this book. In fact both of these most important collections of mathematical reviews credit Mr K-H. Kuhl with zero publication. This raises serious doubts concerning Mr Mr K-H Kuhl expertise in the subject of prime numbers, and concerning the encyclopedic quality of his book. Sapphorain (talk) 11:40, 22 January 2017 (UTC)
 * Given the 2017 publication date, the absence of a review is not nearly as indicative as you seem to think. (Often my research papers do not appear as a listing for weeks, and their reviews sometimes take months.)  Also, this sort of personalization is totally unnecessary, particularly when a perfectly good non-personalized rationale has already been given.--JBL (talk) 15:11, 22 January 2017 (UTC)
 * I do not agree. Even if the book itself is recent, the facts are that (1) the author is credited with zero previous publication on any subject of mathematics, and that (2) this book is a self-publication. Thus I do not agree that the rationale already given is "perfectly good", since it suggests that the book might be used "as a source for this or another article": but this is clearly not so, at least not until it is published by a serious non-paid publishing house. Sapphorain (talk) 16:29, 22 January 2017 (UTC)

using square root of n...
article states

It consists of testing whether n is a multiple of any integer between 2 and the square root of n.

for the number 1,999,999 this is not true... the square root of 1,999,999 is 1414.xxx... but the factors of 1,999,999 are 1657, 71, 17... since 1657 is greater than 1414 then 1,999,999 would be considered prime... but it is not... trust this helps I do not know how to sign this Dan Ellwein2601:3C5:4202:313D:9D48:94A6:1D00:E135 (talk) 05:34, 8 February 2017 (UTC)Dan Ellwein2601:3C5:4202:313D:9D48:94A6:1D00:E135 (talk) 05:34, 8 February 2017 (UTC) — Preceding unsigned comment added by 2601:3C5:4202:313D:9D48:94A6:1D00:E135 (talk) 05:27, 8 February 2017 (UTC)


 * No: since 17 is less than 1414, checking up through 1414 finds the factor 17 and confirms non-primality. (Similarly, 15 has a prime factor larger than sqrt(15), but it necessarily also has a prime factor smaller than sqrt(15).) --JBL (talk) 05:38, 8 February 2017 (UTC)

Natural numbers? Or integers? Negative prime numbers
Why does the article say that primes are natural numbers? I think this is a serious missinformation. Why are the negative primes not included in presentation?--82.79.114.5 (talk) 08:42, 22 February 2017 (UTC)
 * Nearly all definitions either exclude them explicitly or appear to do it implicitly. If they are included then lots of statements about prime numbers have to be reformulated. Prime number includes negative numbers but then they are not called prime numbers. See also https://primes.utm.edu/notes/faq/negative_primes.html. PrimeHunter (talk) 11:00, 22 February 2017 (UTC)
 * I find this very intriguing. Is the reformulation effort so big that it can't be dealed with? --82.79.114.105 (talk) 17:35, 23 February 2017 (UTC)
 * More like, there appears to be no benefit for the cost. —David Eppstein (talk) 18:55, 23 February 2017 (UTC)
 * It may also be a factor that the concept of prime numbers is older than negative numbers. When the latter were introduced there was no good reason to extend the definition of prime numbers. PrimeHunter (talk) 20:03, 23 February 2017 (UTC)
 * The definition of primes is based on the number of divisors which for primes are only improper divisors a,(-a), 1,(-1). Thus the extension is very straightforward. If the extension of the set of divisors with 2 additional improper divisors is not considered, how can then the negatives of natural primes be called? Mirror primes? Negaprimes? I see that antiprime is already taken.--82.79.114.239 (talk) 16:07, 24 February 2017 (UTC)
 * They don't seem important so people haven't found a need to name them. You could describe them like "negative integers whose absolute value is prime" (only two Google hits). PrimeHunter (talk) 16:23, 24 February 2017 (UTC)
 * What's wrong with "negated primes"? —David Eppstein (talk) 17:08, 24 February 2017 (UTC)
 * Of course there is nothing impossible in principle with doing all of number theory with terminology that incorporates the negative primes as primes. (Though really the "correct" thing is to treat the pair {p, -p} or the entire set {np : n integer} as the prime object.)   But there are centuries of habit and reliable sourcing behind the convention on display here, and Wikipedia's role as a tertiary source is to report on existing knowledge, not change existing behavior and convention.  --JBL (talk) 17:43, 24 February 2017 (UTC)

a product of primes that is unique up to ordering
My math isn't great and I'm falling asleep, but the prime number article says:
 * The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

The fundamental theorem of arithmetic article seems to disagree (see the part of this quote I bolded):
 * every integer greater than 1 either is prime itself or is the product of prime numbers, and... this product is unique, up to the order of the factors.

I think the prime number article is missing an important part of that sentence. 3 is an integer greater than 1; if 1 is not a prime number, how can 3 be expressed as a product of primes?

Edit_requests says
 * if you want to make an edit request: Propose a specific change on a talk page. Don't add an edit request template yet.

So... Yeah. Thank you and good night. 71.121.143.38 (talk) 08:10, 30 March 2017 (UTC)


 * In mathematics a "product of primes" can, technically, consist of just one prime - but I agree with you that this usage can be confusing to the general reader. The text from fundamental theorem of arithmetic is clearer, so I have updated the lead paragraph of prime number to use similar wording. Gandalf61 (talk) 09:17, 30 March 2017 (UTC)
 * Mathematicians may sometimes sound odd to laypeople. A "product of primes" can also be the empty product 1 of zero primes so it's often just said that any positive integer can be expressed as a product of primes. PrimeHunter (talk) 11:11, 30 March 2017 (UTC)

About new stuff with prime numbers
I found a great article from the Institute for Advanced Study that I thought might could contribute some to the prime number article!

https://www.ias.edu/ideas/2013/primes-random-matrices — Preceding unsigned comment added by 2001:4642:11C6:0:5D:4668:A89:EB74 (talk) 18:28, 1 July 2017 (UTC)

Prime Number Distribution Series
For a long time, when all the prime numbers up-to some given number were evaluated, it was expected that its 'distribution'/'count off' must/can be represented by a simple analytical function. The distribution of prime numbers is indeed be a pattern related phenomenon but the means that pattern has been sought is misguided/ill-advised, according to Yoldas Askan, a British scientist and mathematician. In his paper, Yoldas challenges some of the fundamental understanding of Prime Numbers and reconsiders these definitions, and ultimately arrives at his analytical formula. In his view, there is no great deal about functions that are approximations because there can be infinitely many of these derived but only suitable at certain number interval. Yoldas claims that the 'beautiful' thing about the Prime number distribution is that there will be no analytical function [of any complexity] that will compute and provide exact values for π(x) other than the Prime Number Distribution Series, which is provided as follows, — Preceding unsigned comment added by Nuclearstrategy7 (talk • contribs) 17:37, 14 July 2017 (UTC)

Proof of the ternary Goldbach conjecture
This has languished on the ArXiv for very long without actually being published – presumably because it is apparently so long that Helfgott's website lists it as to appear as a standalone book. Mind you, if it had not already long since been checked and accepted, one wonders why he got the Humboldt professorship specifically for it. Double sharp (talk) 13:23, 17 July 2017 (UTC)
 * It is not our role to make guesses. The fact is that we have no indication as yet that the paper was accepted in a peer-reviewed journal, and before it is so this information should not be published in an encyclopedia. Sapphorain (talk) 13:36, 17 July 2017 (UTC)
 * Go tell that to Terry Tao and Joseph H. Silverman, then. (The latter even put it on the Goldbach's weak conjecture page back in 2013.) Double sharp (talk) 13:51, 17 July 2017 (UTC)

I have requested comments at Wikipedia talk:WikiProject Mathematics. Double sharp (talk) 14:01, 17 July 2017 (UTC)

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New Solution to the Prime Numbers Problem in Generation and Distribution
There is a new theory is called ASA method, gives a solution to the prime numbers in generation and distribution, Read the published Paper --Alsumery2 (talk) 22:24, 2 September 2017 (UTC)


 * Haven't we seen this before? See Prime Number Distribution Series above. --Bill Cherowitzo (talk) 22:51, 2 September 2017 (UTC)
 * The journal is non-notable and the paper is trivial with an inefficient version of the sieve of Eratosthenes. It's not suited for Wikipedia. PrimeHunter (talk) 23:05, 2 September 2017 (UTC)
 * The journal is worse than non-notable: its publisher, Sciencedomain, was listed by Jeffrey Beall as predatory. As such, nothing it publishes can be treated as a reliable source on Wikipedia. —David Eppstein (talk) 23:36, 2 September 2017 (UTC)
 * The article has been evaluated by numerous reviewers scientists specializing in mathematics and computer science from different universities and from different countries.Many scientific journals and some of which publish research and takes simple fees for publishing, booking and uploading. Evaluation is not a matter of courtesy. Who kills science and ideas who do not care to that researchs to be available for the researchers. Research evaluation does not within a few minutes of publishing when the article is submitted to the free encyclopedia as it happened with me.--Alsumery2 (talk) 14:18, 3 September 2017 (UTC)
 * The author is unknown to MathSciNet, and the journal is not indexed - let alone reviewed - by MathSciNet. This means that neither the author nor the journal is recognized by the mathematical community. Sapphorain (talk) 14:44, 3 September 2017 (UTC)
 * Predatory open access publishing is basically another name for a vanity press. If you've paid to get this published, you can draw your own conclusions. Also, as the others said, it isn't much different from the sieve of Eratosthenes.-- ♦Ian Ma c M♦  (talk to me) 15:18, 3 September 2017 (UTC)

The discussion has become meaningless and sterile Some users are not specialists who are not worthy of the subject-matter evaluations, leaving the subject and going beyond the core of the research is unfortunate Alsumery2 (talk) 15:28, 3 September 2017 (UTC)

Possible typo
In the "Special-purpose algorithms and the largest known prime" section, it says:

"For example, the Lucas' primality test requires the knowledge of the prime factors of n − 1, while the Lucas–Lehmer primality test needs the prime factors of n + 1."

I believe the second link is a typo and should point to some other page. The article for the Lucas-Lehmer test does not suggest that it needs the prime factors of n + 1. Unfortunately, I don't know which article link should be put in its place. The article for the Lucas test does not provide a mention of its n + 1 counterpart. Additionally the MersenneWiki page on primality tests lists only one test that needs the factors of n + 1, calling it "Morrison's Theorem," but the link is dead and I can't find a reference to this theorem online. Derek M (talk) 23:01, 7 January 2018 (UTC)
 * Good call. I agree that the linked article does not support the text here. My suggestion would be to cut off the sentence at the comma, removing the part about Lucas–Lehmer unless or until we find the right link. Additionally there is another much more minor typo: at the start of the sentence, it should either be Lucas' (with the apostrophe) or the Lucas (with the article and no apostrophe) but not the Lucas' (both article and apostrophe). —David Eppstein (talk) 23:21, 7 January 2018 (UTC)
 * Great. I've submitted a request for permission to edit this semi-protected page. If denied, hopefully someone can come along and make the changes I proposed. I've also added the Edit Semi-Protected template to this page. Derek M (talk) 14:06, 8 January 2018 (UTC)
 * Pictogram voting wait.svg Already done Derek M has already made the requested edits. AdA&D  16:18, 8 January 2018 (UTC)