Talk:Prime number

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Former good article nominee Prime number was a good articles nominee, but did not meet the good article criteria at the time. There are suggestions below for improving the article. Once these issues have been addressed, the article can be renominated. Editors may also seek a reassessment of the decision if they believe there was a mistake.
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Deleted[edit]

I struck my own comment as it was useless (due to simple sleight of hand of ref) —Preceding unsigned comment added by Billymac00 (talkcontribs)

The most basic sieve: 6n±1[edit]

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)

@TurilCronburg: 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 Riesel, Hans (1994). Prime numbers and computer methods for factorization. Progress in Mathematics 126 (2nd ed.). Boston, MA: Birkhauser. p. 8. ISBN 0-8176-3743-5. Zbl 0821.11001.  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[edit]

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

Prevalence of primes at a given level[edit]

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]/[[ln(n)]^2], which gives the approximate fraction of prime numbers at a certain 'size' number. For example, at the 1,000,000,000 level, you can expect about 46 out of 1000 numbers to be prime. 71.139.161.9 (talk) 19:37, 7 September 2014 (UTC)