Talk:Prime number/Archive 1

message: On the Prime number page it gives the formula f(n) = n^2 − n + 41 but on the link to the "formulas for primes" page it gives the formula as f(n) = n^2 + n + 41. I don't care to confirm which one is the correct one, or if they both are. Just pointing it out for someone else.
:Both are correct and both are cited by sources. If f(n) = n^2 − n + 41 and g(n) = n^2 + n + 41, then f(n) = g(n-1), so the only difference is a change by 1 in the n values which give primes. PrimeHunter 14:07, 23 February 2007 (UTC)
thread_title: Formula for yielding primes

message: Substituted at 21:56, 3 May 2016 (UTC)
thread_title: Assessment comment

message: I noticed in the "history segment it mentions "the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements (circa 300 BC)" yet in Plato's Theaetetus (lines 147 e - 148 b) which was written in 368 BCE (before even the earliest date Euclid's elements could have been written), It is written that Theaetetus while being taught by Theodorus about square roots he said "we divided number(sic) into two classes" and goes on for a few paragraphs in describing their classification of what he called "Square" and "Oblong" numbers of which had the same function and understanding as what we call "prime" and "composite" numbers. I dont feel the need to put that all in, but I feel it is valuable to mention something along the lines of "Theaetetus, in Plato's Theaetetus recalls solving a problem given by his mentor Theodorus and does so by classifying numbers in Prime-like categories." or something of the sort. BubblesAreAwesome04 (talk) 21:55, 8 January 2019 (UTC)
replies: text: .  Please make a precise request of exactly what text you want to add, and please provide reliable, secondary source(s) to back up any claims.  The "secondary" part here is important.  Drawing any conclusions from original, primary sources would be considered original research and can't be added.  –Deacon Vorbis (carbon • videos) 22:16, 8 January 2019 (UTC)
replies: text: Anyway, I thought Theaetetus was about the theorem that non-square integers have irrational square roots, and therefore about the distinction between square and non-square numbers (which does not really concern prime numbers). —David Eppstein (talk) 02:05, 9 January 2019 (UTC)
replies: text: The relevant text is "For he has discovered a division of numbers into square numbers, 4, 9, 16, etc., which are composed of equal factors, and represent figures which have equal sides, and oblong numbers, 3, 5, 6, 7, etc., which are composed of unequal factors, and represent figures which have unequal sides." 6 is an oblong number without being prime, so the comparison is somewhere between inaccurate and misleading.--♦IanMacM♦ (talk to me) 05:52, 9 January 2019 (UTC)
thread_title:  Plato's Theaetetus as "the earliest surviving records of the explicit study of prime numbers" (1/8/19) 

message: Big Omega function (prime factor) redirects to here, but this article never mentions or defines \Omega(n) much less its asymptotic limit, its summatory function, any Dirichlet series it's involved in, its Lambert series, etc. Did we lose an article somewhere along the way? Not to be confused with little omega: prime omega function. 67.198.37.16 (talk) 03:36, 24 January 2019 (UTC)
replies: text: Never mind, I'll just fix the redirect. 67.198.37.16 (talk) 03:38, 24 January 2019 (UTC)
replies: text:  Good call.  --JBL (talk) 15:33, 24 January 2019 (UTC)
thread_title:  Big Omega? 

message: I read in the history section "Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem ...."  Curious as to how he could have done this several hundred years before Wilson did it, I looked at the Wiki article on Wilson's theorem where I read "This theorem was stated by Ibn al-Haytham (c. 1000 AD)".  In each case the name of our millennial friends and find out that they are the same person.  That was a relief.  Anyways, why is a secondary 'latinized' name used here?  Why not say "the Islamic mathematician Ibn seal-Haytham (often refered to as Alhazen)"?  
secondary point - why say "Islamic"?  I haven't noticed any other mathematician's religion mentioned, unless it is "Islamic".  He was Arab, but again no mention is made of the nationality of Fermat, Goldbach, etc. --2607:FEA8:D5DF:F945:EC2A:FDE3:9E53:A19B (talk) 23:45, 26 January 2020 (UTC)
replies: text: We can and probably should change the link to his name to match the title of his article. As for why Islamic: as an excuse to link to Mathematics in medieval Islam and to set aside this period of development of the theory of prime numbers from the earlier Greek period and the later European period. We do also note other mathematicians as Greek, European, etc in the same section. The name of the medieval Islamic period in mathematics and the sciences has been a matter of considerable debate (with e.g. Persian or Arabic being other contenders) but again, we should match the name of our article on that topic. It is not there as an indicator of the religion of the individual mathematicians (if it were, we should say Muslim rather than Islamic), but as a name for the overall time and place within which they were operating. —David Eppstein (talk) 00:20, 27 January 2020 (UTC)
thread_title:  Alhazen 

message: The section "Infiniteness" should be titled "Infinitude". The primes are infinitely many, but they are not infinite! (That is, each individual prime is not infinitely large.) 2.24.117.60 (talk) 05:00, 16 April 2020 (UTC)
replies: text: I don't think there is a difference in meaning between "infiniteness" and "infinitude", except that "infinitude" sounds both more pretentious and more awkward to me. —David Eppstein (talk) 05:28, 16 April 2020 (UTC)
replies: text: I just changed it, partly to be consistent with the rest of the article.—Anita5192 (talk) 05:43, 16 April 2020 (UTC)
thread_title:  Edit request, 16 April 2020 

message: "A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers."
This sentence and its position in the text indicate this is a complete definition. But it clearly is not, since 2 does not fit this condition and since 1 is a natural number any number greater than 2 does. Taken by itself, this is just a nice way of saying "any number greater than 2".  — Preceding unsigned comment added by Antipattern3 (talk • contribs) 18:50, 23 October 2019 (UTC)
replies: text: I cannot make heads nor tails of your comment.  The given sentence says that 2 and 3 are prime (because neither can be written as a product of two smaller natural numbers) while 4 is not prime (because 4 = 2 * 2 can be written as a product of two smaller natural numbers), etc.  --JBL (talk) 19:14, 23 October 2019 (UTC)
replies: text: This definition is complete and correct. The only possibilities for 2 not to be a prime number would be 1 * 1 = 1, 0 * 0 = 0 and 0 * 1 = 0, which are not 2, therefore 2 is a prime number. 37KZ (talk) 12:32, 12 November 2019 (UTC)
replies: text: This definition initially seemed off to me, too, for 2 - though unlike Antipattern3, not for greater than 2, since I did register "smaller". If you consider natural numbers as starting at 1, it might initially seem discordant because there *aren't* two natural numbers smaller than 2; but of course, the two numbers to be multiplied don't have to be different - 9 is not a prime, because 3 * 3 is 9  — Preceding unsigned comment added by 2601:646:8C81:4E8:F4C2:E493:528E:1F5A (talk) 20:12, 3 May 2020 (UTC)
replies: text: First of all, mathematicians disagree as to what constitutes the natural numbers.  Some include zero; some do not.  One is certainly a natural number.  Second, "two smaller natural numbers" does not necessarily imply "distinct".  For example, four is the product of two smaller natural numbers (2 * 2 = 4), although they are not distinct.—Anita5192 (talk) 22:43, 3 May 2020 (UTC)
replies: text: Whether 0 is considered to be natural is irrelevant to the definition, because it cannot be part of any product showing any other number to be composite. The nonexistence of multiple distinct positive integers smaller than 2 does not somehow magically make it possible to represent 2 as a product of two of them. —David Eppstein (talk) 23:18, 3 May 2020 (UTC)
replies: text: Merriam-Webster says that a prime number is "any integer other than 0 or ± 1 that is not divisible without remainder by any other integers except ± 1 and ± the integer itself." This is closer to the average definition of a prime number than the current opening sentence here.--♦IanMacM♦ (talk to me) 06:55, 4 May 2020 (UTC)
replies: text: We are taking Merriam-Webster as the epitome of mathematics pedagogy? That definition is much less concise and more technical than our lead sentence. It uses division rather than multiplication, a more complicated concept, integers rather than natural numbers, again a more complicated concept, it bizarrely allows −2, −3, −5, etc to count as prime numbers (something that makes sense in abstract algebra but not at the level we want to pitch this article), it has multiple unexplained exceptional cases, it has a negation nested within a negation (not divisible...except), and it uses notation such as ± that is unnecessarily advanced. —David Eppstein (talk) 07:05, 4 May 2020 (UTC)
replies: text: The definition in the opening sentence has changed at some point. In this version of the article from January 2018, it says "A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself." This is very straightforward and doesn't include ± which isn't really needed.--♦IanMacM♦ (talk to me) 07:34, 4 May 2020 (UTC)
replies: text: It is "very straightforward" only if you somehow already know what a divisor is without knowing what a prime number is. I would like the first sentence to be accessible to people who don't. —David Eppstein (talk) 07:42, 4 May 2020 (UTC)
thread_title:  Definition 

message: The article currently says "Fermat primes are primes of the form F_k = 2^{2^k}+1, with k a natural number."
The examples give k = 0,1,2,3 and 4 but the article on natural numbers says that sometimes zero is not included because it is, like the negative numbers, not a number used to count things. To avoid any confusion, wouldn't it be better in this article just to say "Fermat primes are primes of the form F_k = 2^{2^k}+1, with k zero or a positive integer." That has the bonus of making it more obvious to readers that it's the first five cases that are illustrated and that F_5 is the sixth case. While making that change, it might also be appropriate to give the decimal value of F_5 and its factorisation. Michael D. Turnbull (talk) 15:58, 24 May 2020 (UTC)
replies: text: by replacing "natural number" by "nonnegative integer". D.Lazard (talk) 16:14, 24 May 2020 (UTC)
thread_title:  Fermat numbers 

message: "Add to Open Questions at the end of the second paragraph"
Likewise, having as a fundamental value that a compound number is nothing but the sum of a prime number repeatedly, resulting in a multiple of itself or in turn of another prime number that intervenes in the addition, it can be represented that the prime numbers can be formed from additively on a +2n line that has the first prime number as the starting number 3, which is a single additive line that responds to (3+2n). Thus, adding to the number 3 any even number that is not a multiple of 3 results in an odd number that is not a multiple of 3; in the case of the number 5 (3+2*1), the number 7 (3+2*2), the number 11 (3+2*4) and the number 13 (3+2*5) none of the values ​​of n is a multiple of 3, hence the result is not divisible neither by the number 3 nor by any of the numbers previously obtained since the even number that intervenes in the addition to 3 comes from a multiple of 2 that will never have a multiple result of 3, or a lower n value comes from any of the odd results obtained, so they cannot be multiples of them either. Henceforth and given the particularity that any even number multiple of a prime number ceases to be composed of it if the unit or a different prime number is added (proof of Euclid infinite primes), the first result that comes from adding to 3 an even number equal to or greater than a previous prime is also a prime, that is, the number 13 (3+2*5) and thus generically like 17 (3+2*7), 19 (3+​2*8), 23 (3+2*10 which in turn is 3+2[5*2]), the 29 (3+2*13), the 31 (3+2*14 which in turn is 3+2[7*2]), 37 (3+2*17), 41 (3+2*19), 43 (3+2*20 which in turn is 3+2[5*(2*2)]), 47 (3+2*22 which is time is 3+2[11*2]) and it is the number 49 when it is possible to clearly establish the pattern that will continue in the additive line 3+2n the multiple numbers of prime numbers other than 3, which up to this figure (49) it remained "hidden" before the evidence that supposes the odd numbers multiples of 3 (3+2n(3)) and that every number whose unit is 5 is a multiple of this prime number in question.
In the cases of number 9 and number 15, both multiples of 3, it is the case that the multiple involved in the addition itself is a multiple of 3 (3+2*3 and 3+2*6), the latter having a particularity which comes to show the relation that the prime numbers have to each other if they are positioned on a single additive line 3+2n. All multiples of 5, up to 45 fulfill the singularity 3+2[5n+1], as is the already mentioned case of number 15 (3+2*[5+1]) and 45 (3+2[(5*4)+1]). The singularity of 2[5n+1] is repeated in the first multiple that is neither composed of 3 nor of 5. The number 49, which like all squares of a prime number is the first odd number as a product of a prime number that does not it's divisible by all the previous cousins, it responds to 3+2(23), so without taking into account all the aforementioned, double of 23 does not bear any relation to the number 3, but nevertheless, if we attend to a more detailed decomposition, we obtain that 49 is the result of 3+2[(7*3)+2] and it is here that the singularity that relates all the compounds of other prime numbers other than number 3 with the same number 3 is glimpsed in a single addition line 3+2n. This singularity responds to the fact that, in the aforementioned cases, multiples of 5 always result when the even numbers that add up to 3 come from a multiple of 5+1 (2[5n+1]) and will be multiples of 7 when the even numbers that add to 3 come from a multiple of 7+2 (2[7n+2]). What this singularity means (2[(pr*n)+ps]), which is called “position”=ps, is the number of even numbers (in units) that exist between a prime number and the number 3 (that is, its position on the line addition 3+2n).
Therefore, due to the relationship that exists from Euclid's proof with the particularity that any even number multiple of a prime number ceases to be composed of it if the unit or a different prime number is added, in a single line of addition 3+2n that starts with the number 3, prime numbers will result in all those sums in which an even number multiple of 3 (3+2(3n)) does not intervene and likewise primes will also result in all those sums in which it does not intervene an even number multiple of another prime number other than 3 to which its "position" in the additive line 3+2n has also been added, which is obtained from the difference of even numbers (in units) that exist between said prime number and the number 3 (that is, as long as the even number added to the number 3 is not the result of 2[(pr*n)+ps] where ps=(pr-3)/2. Isótopo de Fresa (talk) 10:48, 5 July 2020 (UTC)
replies: text: without any reliable sources reporting on this as an open question; your own original research will not be accepted. Your comment is much too verbose and unclear to even be considered for addition.--Jasper Deng (talk) 11:09, 5 July 2020 (UTC)
thread_title:  Semi-protected edit request on 5 July 2020 

message: The phrase 'natural number' is used several times in this article; I believe 'positive integer' is preferable. Whether the natural numbers contain 0 has been held in contention for years. It is true that, in this context, it does not matter whether the natural numbers contain 0. However, the reader will have to spend time verifying this. Also, 'positive integer' is more common in my experience: see https://books.google.com/ngrams/graph?content=positive+integer%2Cnatural+number&year_start=1800&year_end=2019&corpus=26&smoothing=3. As a result of these considerations, I have changed 'natural number' to 'positive integer' wherever it appears in the article. However, I am open to discuss. — Preceding unsigned comment added by TentativeTypist (talk • contribs) 
:"Natural number" is fine. Including 0 doesn't matter, and the time spent in verifying why is the time spent understanding the definition. XOR'easter (talk) 19:24, 21 September 2020 (UTC)
thread_title:  'Positive integer' vs. 'natural number' 

message: I changed the beginning of the 'Definition and examples' section to this: 'A prime number is a number that is only divisible by 1 and itself. More precisely, an integer p>1 is prime if and only if it has no positive integer divisors other than 1 and p.' The definition I said that was 'more precise' has been criticised for not being more precise. I argue that it is because: (1) it doesn't use the vague word 'number', (2) it specifies that p must be greater than 1, (3) it notes that we are considering positive integer divisors. For instance, 5 does technically have divisors other than 1 and itself, namely -1 and -5. — Preceding unsigned comment added by TentativeTypist (talk • contribs) 
:The previous phrasing (which was almost exactly the text that existed at the time the article gained GA status) was,
::A natural number (1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it is greater than 1 and cannot be written as the product of two smaller natural numbers.
:This already specified the "greater than 1" part, and it excludes negative factors because natural numbers are not negative. Your version is not more precise. In addition, see WP:BRD. XOR'easter (talk) 19:30, 21 September 2020 (UTC)
replies: text: Also see WP:TECHNICAL. In an article with the expected audience of this one (easily likely to include elementary school students), mathematical precision in the lead is less important than readability. The proposed changes are worsenings from that point of view. —David Eppstein (talk) 19:56, 21 September 2020 (UTC)
thread_title:  Prime number precise definition 

message: i suggest to add a formal proof that 1 is not a prime number under https://en.wikipedia.org/wiki/Prime_number#Primality_of_oneHuzaifa abedeen (talk) 05:21, 30 September 2020 (UTC)
replies: text: I think you're missing the point. This statement follows trivially from the definition in the first sentence "A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers", because 1 is not greater than 1. The question is not why 1-not-prime follows from our assumptions, but rather why the "greater than 1" was put into our assumptions in the first place. —David Eppstein (talk) 07:08, 30 September 2020 (UTC)

text: Also, the fact that 1 is not a prime is not a theorem, but a convention that is reflected by the wording choice in the definition of a prime. So no proof can be given. D.Lazard (talk) 08:36, 30 September 2020 (UTC)
replies: text: Sure it can be proved. A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers \Rightarrow every number less than or equal to 1 is not prime \Rightarrow 1 is not prime \Box. The issue is not that a proof is impossible, but that it is trivial and unilluminating. —David Eppstein (talk) 16:46, 30 September 2020 (UTC)
replies: text: 1 is a bit like Pluto, which was once a planet but was demoted to a dwarf planet. The original version of Goldbach's conjecture regards 1 as a prime number, but Christian Goldbach would be told off if he did this today; the current definition says no. It was well into the 20th century before 1 was binned from the list of primes.--♦IanMacM♦ (talk to me) 17:41, 1 October 2020 (UTC)
thread_title:  proof that 1 is not a prime. 

message: Shouldn`t at least Bertrand's postulate be mentioned somewhere in the article?--Kmhkmh (talk) 15:59, 24 February 2021 (UTC)
replies: text: Ok, I mentioned it in the history section, as a step between the conjecture of the prime number theorem and its proof. It is so much weaker than other results now known on the density of primes that I think it is mainly of historical value. —David Eppstein (talk) 19:29, 24 February 2021 (UTC)
replies: text: I think it fits there well, another option might have been to mention it (or similar theorems) under elementary properties. While we have much better theorems with regard to prime distribution, its charm (like Euclid's style infinitude arguments) is in its simplicity and requiring (much) less mathematical background to understand them.--Kmhkmh (talk) 05:53, 25 February 2021 (UTC)
thread_title:  Further well known theorem on primes 

message: Re this edit: it is an unsourced addition, but out of the ten Largest known prime numbers, nine are Mersenne primes. This article says "The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers" but it is unsourced there as well. --♦IanMacM♦ (talk to me) 14:00, 25 February 2021 (UTC)
replies: text: From Largest known prime number: "A list of the 5,000 largest known primes is maintained by Chris K. Caldwell", of which approximately 20 appear to be Mersenne numbers. Is it really accurate to single out 0.4% of the numbers from a list of largest prime numbers and write as if they are the majority, omitting all others? Also, keep in mind that this was recently promoted to Good Article, and I want to keep it that way. We can't do that by piling on unsourced and dubious cruft. —David Eppstein (talk) 17:31, 25 February 2021 (UTC)
thread_title:  Mersenne primes 

message: In the section Definition and examples, please list all the 168 prime numbers up to 1000. Please change "the first 25 prime numbers (all the prime numbers less than 100)" to "the first 168 prime numbers (all the prime numbers less than 1000)" in the third paragraph of the section. Then type all of the prime numbers less than 1000 (from 2 to 997). 💔LasneemaninLet's chat! 21:16, 12 May 2021 (UTC)
replies: text: The first line of that section refers to the article, List of prime numbers, which lists those primes and many more.—Anita5192 (talk) 21:32, 12 May 2021 (UTC)
replies: text: Additionally, there is no good reason for listing so many numbers here — it would throw that section of the article out of balance. I note that the editor who made the initial request has been blocked as a sockpuppet of a banned user. —David Eppstein (talk) 22:09, 12 May 2021 (UTC)
thread_title:  Semi-protected edit request on 12 May 2021 

message: The first line is wrong. It should read: "A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers number greater than 1." 50.206.176.154 (talk) 02:04, 26 September 2021 (UTC)
replies: text: It currently says: "A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers." That is sufficient. If one of the numbers is 1 then the other has to be the number itself and that is not smaller than the number. PrimeHunter (talk) 04:27, 26 September 2021 (UTC)
thread_title:  First line wrong 

message: Add this (an actual pattern to the primes): https://en.wikipedia.org/wiki/Draft:Tobi_Primes. TobiOdeyemi (talk) 13:26, 4 October 2021 (UTC)
replies: text: Not done: Wikipedia does not publish original research. PrimeHunter (talk) 14:02, 4 October 2021 (UTC)
thread_title:  Semi-protected edit request on 4 October 2021 

message: "A prime number is an integer greater than 1 that cannot be expressed as a product of prime numbers."
Is technically correct, though I suppose Wikipedia prefers not describing a subject in terms of itself. --94.191.136.102 (talk) 00:21, 18 December 2021 (UTC)
replies: text: It's clever, but I think this would be too confusing to non-mathematicians, who we want to keep as an audience. (Also finding sources for this variation might not be easy.) —David Eppstein (talk) 00:25, 18 December 2021 (UTC)
thread_title:  Alternative definition 

message: User:Gusfriend has suggested merging NewPGen into this article. I strongly oppose this merge. With no sources, there is no sourced content to merge, and unsourced content should not be included here or elsewhere on Wikipedia. Further, this piece of software is far below the level of significance needed to make it relevant for the main article on prime numbers, a topic with a huge history and many more-important subtopics that could not be included in this already-long article. The removal of the prod on NewPGen, by User:Explicit, was a mistake. It should have been left to die quietly. Now it will have to go to an AfD. —David Eppstein (talk) 16:19, 11 February 2022 (UTC)
replies: text: I agree that NewPGen obviously should not be merged here. I don't see a reason to delete the article itself, though. YMMV. - CRGreathouse (t | c) 17:27, 11 February 2022 (UTC)

text: My merge proposal was in response to the removal of the prod. I am happy to support removal of the merge proposal and the AfD on the original page. Gusfriend (talk) 23:39, 11 February 2022 (UTC)
replies: text: If NewPGen should be merged anywhere, then probably to Primality tests, which may be worth considering (I don't currently have any strong opinion on that, since I didn't try locating sources) Felix QW (talk) 13:36, 12 February 2022 (UTC)
thread_title:  NewPGen 

message: Which are co prime number 2409:4053:211D:A54F:2601:A55D:E1E0:D81E (talk) 11:10, 26 June 2022 (UTC)
replies: text: See Coprime integers. Prime number#Analytic properties calls it relatively prime. Math questions can be asked at Wikipedia:Reference desk/Mathematics. PrimeHunter (talk) 12:43, 26 June 2022 (UTC)
thread_title:  Mathematics