Talk:Chen prime

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[edit] Reciprocals

  • Does anyone know whether the sum of the reciprocals of the Chen primes diverges? 218.133.184.53 04:33, 30 July 2006 (UTC)
The sum of the reciprocals of the Chen primes (1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/23 + 1/29 + 1/31 + 1/37 + 1/41 + 1/47 + 1/53 + 1/59 + ...) diverges. Giftlite 03:03, 31 July 2006 (UTC)
Thanks for reply,Giftlite. How did you know that? 218.133.184.53 03:25, 31 July 2006 (UTC)
It's an unpublished proof. Giftlite 13:01, 31 July 2006 (UTC)
Who proved it? Could we see the work?218.133.184.53 14:00, 31 July 2006 (UTC)
It's not ready for publication, yet. BTW, do you know how to publish on arXiv? Giftlite 02:17, 1 August 2006 (UTC)
I don't know. Did you really prove it? I'm not sure if it's an open question. It might as well be known. BTW, how did you prove it?218.133.184.53 06:48, 2 August 2006 (UTC)
Assuming it is proven, where do you think it should be published for the math community to review? As you already know, WP is not the place for original research. :) Giftlite 13:08, 2 August 2006 (UTC)
I don't know whether your result is new or not. If you think you really proved it, paste on this page. Who cares!218.133.184.53 19:00, 2 August 2006 (UTC)

Is there a reference from a reputable source about whether the sum of reciprocals converges? Or a trivial proof?
There are more Chen primes than twin primes, so convergence doesn't follow from Brun's theorem.
Here is PARI/GP code computing the sum at the first powers of 10:
(22:27) gp > s=0;for(n=1,7,forprime(p=10^(n-1),10^n,\
if(bigomega(p+2)<=2,s+=1.0/p));print("10^"n": "s))
10^1: 1.17619
10^2: 1.72650
10^3: 1.99857
10^4: 2.14398
10^5: 2.24302
10^6: 2.31244
10^7: 2.36492
This data does not prove anything. If somebody wants to add a claim to the article then please provide evidence here and wait for replies. I haven't seen evidence to claim convergence or divergence.
To 218.133.184.53: If you add a claim to the article then you have the burden of proof.
If you don't show proper evidence then another editor can delete without having to prove your claim is false. PrimeHunter 21:03, 7 August 2006 (UTC)

Thank you, PrimeHunter. BTW, do you know anyone who can give me the permission to publish on arXiv? Giftlite 23:02, 7 August 2006 (UTC)
No. If you haven't tried already then look here and try filling in the author registration form. PrimeHunter 00:45, 8 August 2006 (UTC)

[edit] is this for real?

is this for real? is their really such a thing as chen primes? list of prime nums lists the chen primes as 2,3,5,7,11,13,17,19,23,29,31,37,41,47. sloane's oeis has just two matches, oeis:A078133 Primes which cannot be written as sum of cubes>1. and oeis:A086472 Primes which are sum of two palindromes.

  • Yes, they are real. The above two sequences are not Chen primes. Therefore, OEIS is incomplete.
Then someone needs to send in to the OEIS. Because if I was a deletionist, the lack of an OEIS entry would be excuse enough for me to stick a VfD tag on any mathematical article. Anton Mravcek 18:36, 30 July 2005 (UTC)
I want to support this article, but right now it is deletionist bait. There is no OEIS entry and no MathWorld entry either (though there is an entry for Chen's theorem, and that involves semiprimes). A Google search for "Chen prime" turns up a Wikipedia page as the first result, but almost all the other results are unrelated to number theory (e.g., "candidate Lien Chan ( 連戰 ) will be Chen's prime target," from the Taipei Times). PrimeFan 17:47, 1 August 2005 (UTC)
I dislike deletionists who act before they think. Worse, sometimes they delete to further their own cause. Left uncorrected, they can do more harm than good. I also dislike decisions by committees. Most important math discoveries are made by creative individuals, not committees. In order to support this article, however, I've e-mailed the Chen primes to OEIS. Giftlite 03:04, 6 August 2005 (UTC)
I have verified that the professional mathematical paper mentioned in the article does indeed contain the term Chen prime, in Theorem 1.2 on page 5. This together with the OEIS entry convinces me of the validity of the term. Anton Mravcek 20:36, 6 August 2005 (UTC)
Has Chen prime been used anywhere but that paper? In the paper, Chen prime is defined for the first time in theorem 1.2, and we're reminded of the definition twice afterward. Then in section 6, the authors redefine Chen prime (including the condition that both factors of p+2 exceed p^3/11) and use this definition henceforth. All of this suggests to me that Chen prime is more like a nonce piece of terminology invented for that paper than an accepted name for these primes.
On another note, as long as we're submitting sequences to the OEIS, we might also want the non-Chen primes (43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, 271, 277, 283, 313, 331, 349, 367, 373, 383, 397, 421, 433, 439, 457, 463, 523, 547, 593, 601, 607, 613, 619, 643, 661, 673, 691, 709, 727, 733, 739, 757, 773, 823, 853, 859, 883, 907, 929, 967, 997, 1013, 1021, ...) of which there are fewer among small integers. 4pq1injbok 01:17, 8 August 2005 (UTC)
The non-Chen primes are in the OEIS now, A102540. PrimeFan 20:31, 22 August 2005 (UTC)
Mm, good point. My next question is, then, what is the standing of the two mathematicians involved? Is it likely that other mathematicians will accept the term and use it in future papers? Anton Mravcek 19:41, 10 August 2005 (UTC)
In answer to my own question, Neil Sloane has accepted the term for use in the OEIS. A109611 gives the name "Chen prime" without quotation marks. Anton Mravcek 23:34, 18 August 2005 (UTC)
The issue remains about the redefinition of "Chen prime" to include "the condition that both factors of p + 2 exceed p^3/11" that 4pq1injbok. I will have to carefully read the paper. PrimeFan 20:31, 22 August 2005 (UTC)

[edit] Why?

I removed the statement about an arbitrary decimal made by concatenating Chen primes. I have three reasons for this.

1. The proposition is of no mathematical interest. Write it as a tortuous series in powers of 10 to products of integers parts of logs to base 10 of Chen primes and the only thing you should think is "why?"

2. The statement was merely one of ignorance about the proposition of no mathematical interest.

3. The proposition is obviously true given the existence of an infinite number of Chen primes, but not worth the effort to check (think about it - is it at all plausible that the decimal could recur?) Elroch 20:45, 8 March 2006 (UTC)

I'm glad this article is gaining more interest. I think we better let the mathematical community decide whether this statement is interesting or not. Why don't we vote on this? Giftlite 00:33, 9 March 2006 (UTC)
In hindsight it was reckless to say anything is of no mathematical interest. Erdos and Copeland were interested enough to show that the (aptly named) Copeland-Erdos constant was normal. This isn't a big surprise, as the set of non-normal numbers is of measure zero. What does look of interest are the continued fractions for the Champernowne constants. These constants have tidy looking expressions as series, which intuitively makes them more likely to exhibit some sort of structure, but the weirdness of their continued fractions is a phenomenon which I am not sure is fully understood.
However, the main idea of an encyclopedia is to collect things that are known, with the only exception being things that people would really like to know, and have tried hard to know, but have failed. The deleted proposition does not fall into this class. Elroch 00:39, 9 March 2006 (UTC)

Is it worth writing if I proved the irrationality?WAREL 00:52, 9 March 2006 (UTC)

I wish you well with that laudable ambition. You would need to pass the WP:NOR criteria of course, to refer to it in an article. Hopefully, you understand by now what that entails. Elroch 01:04, 9 March 2006 (UTC)
Right. Please provide references. WAREL, you are a smart chap. You should go to the math department, not economics. Then you can publish papers in math journals rather than trolling around on Wikipedia. Oleg Alexandrov (talk) 01:32, 9 March 2006 (UTC)


[edit] Aside

In looking at the Highly Composite Numbers(HCN)OEIS A2182, I note that the prime number nearest to each is often a Chen prime, at least as far as I have the data.--Billymac00 02:30, 24 April 2006 (UTC)

Interesting. By "nearest" do you mean - 1 or + 1? Or both? Anton Mravcek 21:35, 24 April 2006 (UTC)
I believe by "nearest" he means the prime whose difference with the number in question has the smallest absolute value. OmnipotentEntity 20:02, 15 June 2006 (UTC)

[edit] TG prime

Terence Tao and Ben Green proved in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes. A Chen prime that could be at the second of the progression is called TG prime. For example, 5 is TG prime because (3,5,7) is a three-term progression of Chen primes. Accordingly, there are infinitely many TG primes.

The first few TG primes are

5, 7, 11, 13, 17, 19, 23, 29,31), 37, 41, 47, 53, 59, 67, 71, 83, 89, 101

—The preceding unsigned comment was added by 218.133.184.93 (talk • contribs) 02:11, January 21, 2007 (UTC)

"TG primes" are unsourced. I asked for a source on User_talk:218.133.184.93 with no reply. PrimeHunter 18:50, 21 January 2007 (UTC)
I put the listed "TG primes" in the OEIS search box and the only result that came up was the Chen primes. Some mathematical concept will someday bear Tao and Green's names. I think it'll be something other than this. Anton Mravcek 22:45, 22 January 2007 (UTC)
I wonder who Green-Tao theorem is named after ;-) PrimeHunter 12:53, 23 January 2007 (UTC)

[edit] External link removal

Someone just removed an external link to a site about some largest known Chen prime. I think the article doesn't read right without this link now. If an admin out there agrees with me, please replace the link. Giftlite 17:25, 21 March 2007 (UTC)

I am Jens Kruse Andersen. I organized the search for that Chen prime record and wrote the announcement http://groups.yahoo.com/group/primeform/message/6481 which is the removed link. Before registering at Wikipedia and reading guidelines, I added the record and external link (diff). The link is a mail list post which is often frowned upon in Wikipedia. Maybe it affects reliability concerns that it's currently referenced in http://Mathworld.wolfram.com/ChenPrime.html, http://primes.utm.edu/primes/page.php?id=75857#comments and http://primes.utm.edu/bios/page.php?id=339. Note that the MathWorld article was written by me (with minor and undocumented changes by the editor), so maybe the template should say {{MathWorld | urlname=ChenPrime | title=Chen Prime| author=Jens Kruse Andersen}}. I'm not an admin and don't care whether the link stays out or gets back in. The MathWorld article (although having the same author) is a more reliable source and mentions the record. However, it doesn't mention Micha Fleuren's involvement, so if the link stays out then maybe http://primes.utm.edu/primes/page.php?id=75857 or http://primes.utm.edu/bios/page.php?id=339 should be referenced. I will leave the decision to others due to WP:COI. PrimeHunter 14:30, 22 March 2007 (UTC)
Generally, I'd say Wikipedia shouldn't use links to Yahoo! Groups. In the case of articles about porn stars, the links tend to be to fan clubs and are usually quite un-enlightening. "Tara is better than Belladonna!" "No way..."
But it can be a dangerous thing to make blanket assumptions. When it comes to Web forums about mathematics, and the particular Yahoo! Group in question, there is an interest in pointing out mistakes, and in not taking things at face value. For example, the reply from Pierre Cami shows that he checked whether your discovery meshed with his computer's data.
So in conclusion, to the others, I say that I think this particular Yahoo! Group link is worth having in the article. Anton Mravcek 18:59, 22 March 2007 (UTC)