# Talk:Prime number

Although involving products in the definition instead of divisors is correct (and rather elegant), it is still pompously so. It requires a non-negligible amount of thinking for the uninitiated, and I have never seen it worded as such anywhere else, nor do math profs usually teach it that way.
This very clearly violates WP:ASTONISH by trying to be clever, besides obviously being WP:OR. 105.156.135.60 (talk) 19:49, 26 September 2021 (UTC)

You have got to be kidding me. It is far from being original research – sources for this definition are not difficult to find – and in fact is deliberately non-technical by using multiplication, a simpler and more familiar concept than that of a divisor. In the US, students start learning about multiplication in the second grade. Division is a third-grade concept, and factorization, multiples, and divisors are fourth-grade concepts. When it is possible to express things more simply using lower-level concepts, that is what we should do. —David Eppstein (talk) 20:13, 26 September 2021 (UTC)
Googling "definition of prime number", and going as far as four pages deep, the one and only website that mentions a product —mathisfun.com— does so by avoiding the need from the reader to realize that "two smaller factors" means 1 is not included (by stating so explicitly), since that would involve the number itself (which is not smaller). And as you can see in the talk page again and again, and presumably why the article is restricted, probable adults are still getting confused by the wording, and I guarantee you that 2nd, 3rd, 4th graders and the rest of the population will be too, with a likelihood far more significant than that of 2nd graders reading about prime numbers on Wikipedia while still being unaware of the concept of divisors.
I write code for a living, and I sure as heck love my clever one-liners, but my coworkers and other people that have to understand/maintain my code don’t really. As such, I tend to avoid them. I hope you will do the right thing here. 105.156.135.60 (talk) 21:35, 26 September 2021 (UTC)
Resisting the temptation to write a clever one-liner about programmers or about your reliance on web searches instead of reliable sources, a quick search quickly found examples of sources that define it in exactly this way: [1] [2] [3]. —David Eppstein (talk) 21:50, 26 September 2021 (UTC)
(edit conflict)I guess that the definition that you find less astonishing is the one that suppose to know what is a divisor. Wikipedia is aimed for the widest possible audience. So a definition that use less technical concepts (here the concept of divisor) is less astonishing and more adapted to a large audience. It is not the fault of Wikipedia if some teachers use wordings that are less simple than needed. D.Lazard (talk) 20:17, 26 September 2021 (UTC)
Surely you mean "more complex than needed"??? :P EEng 03:33, 25 March 2023 (UTC)
If I have 4 apples and divide them equally and whole to 2 teachers it is no stretch to consider: 2 apples each. How is this any different than 2 apples and 2 teachers if 1 apple each? First example: 4 apples Teacher 1 gets 2 Teacher 2 gets 2, Second example: 2 apples Teacher 1 gets 1 Teacher 2 gets 1. The division is whole and distribution is equal in both cases. Please note I am so sorry to raise the hoop and jump through it for nobody objecting yet. But, this is neither original research nor clever it is a simple truth! If not on Wiki, where must reason rule? 50.220.179.34 (talk) 07:42, 28 October 2023 (UTC)
i also second that the definition used here is not very canon. For example, i checked the German, the French, the Italian, and the Spanish Wiki, and they all use the definition based on having two divisors. Maybe its a US-american thing to use the definition based on factors, but its definitely not commonly taught like that in Europe. The definition used here even is based on a negation ("any number that is not ...."), which is didactically not very clever.  That said, even if the factor-based definition is to be kept, a much better version would be "that have exactly two factors, themselves and 1", as suggested by the BBC. The German Wiki uses the word "divisor" instead of "factor", and this is arguably the best way to define a prime. sorry for necro'ing this discussion. You will likely not change it anyway, because someone owns this place :> 2A00:6020:4504:AF00:301A:5A8B:DEDE:22B5 (talk) 22:50, 10 June 2024 (UTC)
Agree. This is a problem. The definition in the article summary ("Number divisible only by 1 and itself") is clearly the simplest to understand. It should be the very first line of this article. Instead the intro jumps straight into maths jargon like "product", which to ordinary people is something found in a supermarket. Needs rewriting. Rollo (talk) 10:45, 3 September 2024 (UTC)
Huh! My interpretation of this is correct, technically. Nonetheless, the "Mathematical Olympiad" source (yet wrong page, should be p. 25) says the definition of a prime number is a natural number greater than 1 that cannot be factorized as the product of smaller natural numbers. It is also reasonable, but I think people may not get used to the alternative definition.
Forget about this! It is nothing but an article summary. Dedhert.Jr (talk) 12:23, 3 September 2024 (UTC)

This puzzles me: two apples, two teachers, one each. Four apples, two teachers, two each. How then the prime? Would not the best be divided into equal whole equal parts  ? 50.220.179.34 (talk) 05:56, 6 June 2023 (UTC)

Can you clarify what you mean by that? If you are simply saying that a set of two objects can be partitioned into two sets of one, then of course that is true, but so what? What does that have to do with content of the article Prime number? If, on the othe hand, you don't mean that, then I have no idea what you do mean. JBW (talk) 12:59, 6 June 2023 (UTC)
trolling? Dhrm77 (talk) 13:54, 6 June 2023 (UTC)
Two is the only even prime number. Beyond that, it's hard to say what this question means.--♦IanMacM♦ (talk to me) 14:53, 6 June 2023 (UTC)
I am sorry this was also addressed here but with a little less detail. (I top posted). I believe 2 must not be prime because it is both a whole and equally portion-able number. I thought composite meant 'can share equally and whole' (for both school-aged/elementary and math aged-giants). 50.220.179.34 (talk) 07:54, 28 October 2023 (UTC)
It seems that you use "equally portion-able number" in the sense of even number. It is true that 2 is an even number, and that it is the only even number that is also a prime number. Please, be care that using current English words instead of accurately defined mathematical words can be the source of many mistakes. In particular your thought about the mathematical meaning of "composite" is erroneous: the definition of a composite number is to be a whole number that can be obtained by multiplying two smaller whole numbers. This is not related with 'can share equally and whole', whichever meaning is attributed to this phrase that is nonsensical for me. D.Lazard (talk) 09:01, 28 October 2023 (UTC)
What is meant by "equally portion-able number" even as say 77 can be 7 of 11 or 11 of 7 and whole numbers so can be odd. This makes sense to me. 50.220.179.34 (talk) 05:14, 1 November 2023 (UTC)
Every positive integer n may be divided into n copies of 1 or into one copy of n; one might call these the "trivial factorizations".  A number is composite if it also has some other ("nontrivial") decomposition into an equal number of equal integers (equivalently, in which both the number of pieces and the size of the pieces is smaller than the number n itself).  Four apples may be divided into equal piles in three different ways: four piles each containing one apple (trivial), one pile containing all four apples (trivial), or two piles each with two apples (nontrivial).  --JBL (talk) 19:56, 15 November 2023 (UTC)

I don't know why, but since my last edit on this article 30 minutes ago, I can't edit this article anymore. It always says "The server did not respond within the expected time.", no matter how often I try to publish my changes. It's so weird because I can edit other articles just fine. I already tried restarting my device and to re-login. Nothing works. Please help, thanks.-- Maxeto0910 (talk) 22:58, 20 July 2023 (UTC)

Based on the version history, it seems like my edit was now published after trying it dozens of times, though they still aren't displayed in the article for me. I never experienced those problems since editing Wikipedia.-- Maxeto0910 (talk) 23:26, 20 July 2023 (UTC)
This seems like a Wikipedia technical problem. It's not about you. –jacobolus (t) 00:14, 21 July 2023 (UTC)
I cannot even read the article, the server blocks on this particular page.--Sapphorain (talk) 07:21, 21 July 2023 (UTC)
I have been seeing slowdowns for a few days on many long Wikipedia mathematics articles. See discussion of the same issue for golden ratio at Wikipedia:Help desk § Golden ratio. —David Eppstein (talk) 07:26, 21 July 2023 (UTC)

We have a problem! The definition of a prime number in the first sentence is wrong. It currently reads "A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers." By this definition 8 is prime. It should read, "A prime number (or a prime) is a natural number greater than 1 that is not a product of smaller natural numbers." Or, "A prime number (or a prime) is a natural number greater than 1 that is not a product of two or more smaller natural numbers."  — Preceding unsigned comment added by 50.206.176.154 (talk) 20:43, 10 November 2023 (UTC) 

The number 8 is the product of 2 · 4, and is therefore a composite number. –jacobolus (t) 21:15, 10 November 2023 (UTC)
It says a natural number greater than 1 that is not the product of two smaller natural numbers. Dedhert.Jr (talk) 12:27, 3 September 2024 (UTC)
Right, and hence, 8 is not a prime number. - CRGreathouse (t | c) 16:35, 3 September 2024 (UTC)
If that is the case 2 (Natural and smaller) + 3 (Natural and smaller) = 5 therefore 5 can't be a prime number and instead a composite number. The proper definition of Prime should be; a natural number that isn't 1 (or greater than 1) that can only be divided by itself and 1. 201.216.219.193 (talk) 17:44, 1 July 2025 (UTC)
You are adding two numbers. The definition involves multiplying two numbers. Adding is not the same as multiplying. —David Eppstein (talk) 18:02, 1 July 2025 (UTC)

there are multiple prime numbers. this article is about all of them. Rguyr (talk) 17:21, 30 December 2023 (UTC)

Cf. WP:SINGULAR and WP:PLURAL. –jacobolus (t) 18:03, 30 December 2023 (UTC)
I agree that would be a good idea 71.14.253.152 (talk) 15:02, 14 March 2024 (UTC)
If it's common and reasonable to talk about one prime number at a time (as it is), then Wikipedia's standard convention is to use a singular title. —David Eppstein (talk) 17:52, 14 March 2024 (UTC)

 Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. - FlightTime (open channel) 15:55, 3 May 2024 (UTC)

Subject says it all. Cannot edit it myself as it is locked. RonanTetsu (talk) 01:01, 20 September 2024 (UTC)

 Done —C.Fred (talk) 01:02, 20 September 2024 (UTC)

The definition refers to 1 and itself, you cannot divide 1 number, you need 2 numbers. Even if the value of both numbers are 1, it is still two numbers, never itself. ZhuLien (talk) 09:51, 31 October 2024 (UTC)

Nice explanation here.--♦IanMacM♦ (talk to me) 10:17, 31 October 2024 (UTC)
exactly. 1 and itself is different than two numbers each with the value 1 ZhuLien (talk) 08:01, 1 November 2024 (UTC)

Rule of 24: If P is a prime number > 3, then P^2 = 24K + 1, where K is an integer.  
This is not a conjecture and can easily be proven:

    Let P = 2n + 1
    P^2 = 4n^2 + 4n + 1 
    P^2 - 1 = 4n(n+1) 

thus P^2 - 1 is divisible by 8 since n(n+1) must include an even number.  In addition n or n+1 must be divisible by 3 since if not then n+2 would be divisible by 3.  If so, then 2(n+2) = 2n + 4 would also be divisible by 3 as well as 2n+4 - 3 = 2n + 1.  But 2n + 1 represents the prime number P, so n + 2 cannot be divisible by 3.  Thus n or n+1 must be divisible by 3.  Therefore P^2 - 1 must be divisible by 24 and the Rule of 24 is proven. DrClark3 (talk) 20:01, 18 June 2025 (UTC)

We cannot include material like this without published reliable sources. See also WP:NOR. —David Eppstein (talk) 20:09, 18 June 2025 (UTC)
Moreover, numbers ⁠
  
    
      
        P
      
    
    {\displaystyle P}
  
⁠ such that ⁠
  
    
      
        
          P
          
            2
          
        
        =
        24
        K
        +
        1
      
    
    {\displaystyle P^{2}=24K+1}
  
⁠ are exactly the odd numbers that are not mutiple of 3, and this can be tested without squaring them by looking to the last digit and to the sum of their digits. D.Lazard (talk) 09:12, 19 June 2025 (UTC)
For the reference, it's difficult to pinpoint the first reference but Carl Friedrich Gauss does include it in "Disquisitiones Arithmeticae" in 1801.  Most likely Fermat and Euler also knew it but did not explicitly state it.
I may be missing the point about the multiples of 3 comment, but the Rule of 24 doesn't claim that P^2 is a multiple of 3, but rather P^2 - 1, which is a multiple of 3 (and 8) as show in the proof.  I note that I have found other more elegant ways to prove the Rule of 24, such as:
P^2 - 1 = (P+1)(P-1) since P is prime >3 either P+1 or P-1 must be a multiple of 3
And, as shown above: P^2 - 1 = 4n(n+1) is a multiple of 8. 2605:A601:A6A7:2100:CC1:E9C3:811E:F2B7 (talk) 21:01, 10 October 2025 (UTC)
Nobody said anything about ⁠
  
    
      
        
          P
          
            2
          
        
      
    
    {\displaystyle P^{2}}
  
⁠ being a multiple of 3. What was said is that ⁠
  
    
      
        
          P
          
            2
          
        
        =
        24
        K
        +
        1
      
    
    {\displaystyle P^{2}=24K+1}
  
⁠ if and only if ⁠
  
    
      
        P
      
    
    {\displaystyle P}
  
⁠ is an odd number that is not a multiple of 3. (Which is very easy to prove.) Therefore what you call the "Rule of 24" is no more than a roundabout way of saying that if ⁠
  
    
      
        P
      
    
    {\displaystyle P}
  
⁠ is a prime number > 3 then ⁠
  
    
      
        P
      
    
    {\displaystyle P}
  
⁠ is odd and not a multiple of three; that is trivially true. (Incidentally, the proofs you give make no use at all of the fact that P is prime; they use only the facts that P is odd and not a multiple of 3.) JBW (talk) 00:55, 12 October 2025 (UTC)

There is a new discovery described in article:

Crowell, Rachel. "Mathematicians Come Up with 'Mind-Blowing' Method for Defining Prime Numbers". Scientific American. Retrieved 2025-08-15.
About a new way of finding if a number is a prime using special Diophantine equation and partition functions.
I wanted to include the information, but not sure where to put it.
jcubic (talk) 20:05, 15 August 2025 (UTC)