# Talk:Twin prime

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## New record twin prime?

On 8 September 2005, 84.130.217.208 updated the "largest known twin prime" here and on Chen prime to record a new record. I've not been able to find any reference to this new record via Google, so I've asked for one on the contributor's talk page, but in the meantime if anyone knows anything about it please shout. Hv 19:40, 8 September 2005 (UTC)

• I've got good news and bad news. Good news: 1. The entry is correct, 2. I'm delighted the largest known Chen prime is also found, and many people have found Chen primes very interesting. Bad news: We don't know who updated the Twin prime and Chen prime pages to announce this good news. Giftlite 23:28, 8 September 2005 (UTC)
Thanks; I'm kind of surprised there isn't anything found by Google on this, but I guess I shouldn't be. Hv 01:18, 9 September 2005 (UTC)
could have been me. and i just updated to a new alrger prime, though not much larger. they always come here http://primes.utm.edu/top20/page.php?id=1 and are anounced in some yahoo groups. all are verified primes. -thommy

## Factorial

Why is factorial under see also? Ozone 03:13, 2 January 2006 (UTC)

I think because this operator appears in this equation: ${\displaystyle 4((m-1)!+1)=-m\mod (m(m+2))}$. Giftlite 02:27, 1 February 2006 (UTC)
As to that formula, apart from changing the depreciated "proven" to "proved" (according to wiktionary), it would be nice to have a reference for that proof. — MFH:Talk 02:42, 25 March 2008 (UTC)
It looks like this is just a trivial consequence of Wilson's Theorem and the Chinese Remainder Theorem, both from elementary number theory; it's an exercise, and the proof is a few lines. Of course the proof doesn't belong here, but maybe we can rephrase "it has been proven that" to "it can be proved that" or possibly "it is easy to show that". One other thing, I would also recommend that this statement and the one after it be moved to the "properties" section of this article, if kept at all. I won't edit it yet, I guess, because I want to see whether or not others think these entries are worth keeping. Kimbesque (talk) 02:21, 15 May 2013 (UTC)

## Mathematical Definition of Silly

Regarding this edit:

02:53, 8 March 2006 Dmharvey m (remove silly "proof" from external links -- thanks again DYLAN)


What exactly is the definition of 'silly'? Also, while you're at it, please define mathematically 'arrogance'. These are external links, not legal testimony. Please stop squelching any innovation that doesn't fit in to your narrow minded view of mathematics. If you feel you are mathematics' custodian, please fashion yourself a thrown to sit on, only make sure it's located somewhere out on the Atlantic Ocean where we needn't listen to people such as yourself. —The preceding unsigned comment was added by 38.116.204.67 (talkcontribs) 18:46 and 18:51, 13 March 2006 (UTC)

## Merge with Twin prime conjecture?

OK, it's possible to discuss the two topics separately, but is it useful? It's not as though we're hurting for space; the two articles combined come to less than 10K. (BTW there's an inconsistency; twin prime says a twin prime is an individual prime, whereas twin prime conjecture says it's a pair). --Trovatore 21:15, 13 March 2006 (UTC)

I agree. I don't see a good reason to keep the articles separate. The topics are quite intertwined, the twin prime article is barely over stub length, and the contents of the articles overlap. (I fixed the inconsistency.) -- EJ 21:47, 13 March 2006 (UTC)
I agree also. Radagast3 (talk) 05:05, 22 March 2009 (UTC)
Help:Merging and moving pages#Proposing a merger describes the procedure to propose a merger. PrimeHunter (talk) 17:48, 22 March 2009 (UTC)
I agree, and I have added merge proposal templates to both pages. Jim (talk) 04:06, 13 April 2009 (UTC)
Seems like a good idea to me. —David Eppstein (talk) 04:19, 13 April 2009 (UTC)
It's been several months since the templates were added. I think it's time to begin the merge. --joeOnSunset (talk) 22:30, 27 December 2009 (UTC)

Since several people have supported this merge and none opposed over nearly 3 years I have done it. JamesBWatson (talk) 14:58, 13 January 2010 (UTC)

## Merge with Twin prime conjecture - They're equally stubborn

There are equally self righteous mathematicians goal tending that page. The EXTERNAL LINKS section is intended for EXTERNAL LINKS, not only for published mathematical papers. 38.116.204.67 21:27, 13 March 2006 (UTC)

They are not for published papers, the "references" section is for that.— MFH:Talk 02:42, 25 March 2008 (UTC)
It's not for links to just anything. There's got to be some quality control, or we'll be overrun with links to crackpot sites. I'm not saying that's what your site is; I haven't looked into it that closely. (My first guess is that you may very well have a good, convincing argument in favor of TPC, but you probably don't have an actual proof, and you may not fully understand the difference.) In any case I'd say there's a "rebuttable presumption" against amateur sites. Feel free to have a shot at rebutting it. A good start would be to convince a recognized expert. --Trovatore 21:35, 13 March 2006 (UTC)
Here's a great question for you. Suppose a 32 year old computer scientist has an insight. Suppose that the insight is ground breaking. Suppose that any local expert the computer scientist approaches lacks the insight to see the deeper meanings. Suppose the computer scientist needs to reach a 1 in a million expert. Now suppose that your stubborn refusal blocks this process until long after the computer scientist is dead, and as a result, we never get warp engines, clean power supplies and all those other things that would make the world a nicer place? Think I'm nuts? Look at the connections between the riemann(sp?) and quantum mechanics. Enjoy your particular mountain for as long as you are king of it. —The preceding unsigned comment was added by 72.136.90.95 (talkcontribs) 03:52, 14 March 2006 (UTC)
Independently of any question about insight, there is a basic principle on WP which is called "no original research". This means that you must indeed publish your results elsewhere, and even after this wait until it is "commonly accepted", before putting it here. Feel free to leave a link on this talk page, however, if you feel the urge. — MFH:Talk 02:42, 25 March 2008 (UTC)
I assume you are a 32-year old computer scientist. Have you actually tried asking any "local experts"? (By the way, you might be taken more seriously here if you create an account, and sign your messages with four tildes: ~~~~. It also makes it easier to keep track of the discussion. Thanks.) Dmharvey 14:38, 14 March 2006 (UTC)
But of course. My professor that introduced me to this problem is super busy with no time to talk to me. You see there's nothing in it for them (experts). Suppose I'm right: then all they do is help some other person get credit for a big discovery. Suppose I'm wrong: they've wasted their time. Either way they've done nothing for themselves. It's much easier to dismiss me. I'm trying to find someone who has an open mind for these things. It's very difficult to find. 38.116.204.67 17:24, 14 March 2006 (UTC)
Good God, is this what goes on in Mathematics circles outside Humanities faculties? In that case thank goodness I pursued a pure mathematics course because I always found my professors incredibly tolerant of my supposed insights! Just a suggestion, but perhaps they might listen to you more if you couched your brilliant breakthrough in humbler terms? One gets the impression that our friends in the impure sciences :-p seem to forget that ultimately all research is a human endeavour and relies on human interaction at some level to enable progress, and mutual affability expedites the immediate openness of others to one's ideas.Gondooley 11:50, 20 May 2006 (UTC)

## Brun's constant

http://en.wikipedia.org/wiki/Brun's_constant

Prime Twin Constant called the Brun's Constant ! and not 1.3xxxxx

thx

Patrick Bertsch —Preceding unsigned comment added by 134.60.222.1 (talkcontribs) 20:43, January 7, 2007

AFAICS, it's neither that nor the other: it's twice the twin prime constant 0.6601..., cf. Twin prime conjecture (after correcting the product which should run over (all, not only twin-) primes p>3). To get Brun's constant, 1.9..., you sum over twin primes only.— MFH:Talk 02:42, 25 March 2008 (UTC)
As to that formula, I'll add that p must be prime, and from changing the depreciated "proven" to "proved" (I'll fix these two), it would be nice to have a reference for that proof.[cf "Factorial" above] — MFH:Talk 02:42, 25 March 2008 (UTC)

## Pairs that are being discovered

Most of the pairs that is being discovered are of the form (4m-1,4m+1). However, the proof showing there are infinitely many twin primes is not enough to show that there are many pairs of the form (4m-1,4m+1).　It also has not been shown whether the highest pair is of the form (4m+1,4m+3) or (4m-1,4m+1) if there are only finitely many twin primes. 218.133.184.93 19:13, 28 January 2007 (UTC)

No "proof", no reference to a proof. It doesn't belong in the article. — Arthur Rubin | (talk) 18:13, 28 January 2007 (UTC)
Even rather counter-evidence: e.g. in 1e7..2e7 and 2e7..3e7 there are more twins (p,p+2) with p=1 (mod 4) than with p=3. The fact that the largest known twins are of the form k*2^n +/- 1 of course stems from the fact that only numbers of that form have been considered! — MFH:Talk 04:27, 25 March 2008 (UTC)

## Inconsistency?

I'm a little confused by the following two sections:

the number of twin primes less than x is << x/(log x)2.
the number of such pairs less than x is x·f(x)/(log x)2 where f(x) is about 1.7 for small x and decreases to about 1.3 as x tends to infinity.

They seem contradictory to me. Can anyone explain this better either to me or on the page?99of9 (talk) 05:05, 19 February 2008 (UTC)

The explanation is simply that the former statement was in error. Fixed now. -- EJ (talk) 12:56, 19 February 2008 (UTC)
Thanks for fixing. 99of9 (talk) 23:31, 19 February 2008 (UTC)

## Dots in the function statement intended to end the sentence

I'm going to repost my argument from Talk:Collatz conjecture instead of writing the same thing over again.

I see that they're *intended* to end the sentences, but they end up confusing the function declaration. I don't think it's necessary to insert the period - the end of the sentence is implicit by the colon before the declaration and the declaration itself. The periods wind up appearing to be part of the mathematical or logical statement and merely confuse things without making anything more clear. I think we should take them out. Kyle Barbour 07:53, 5 April 2008 (UTC)

O.K. After a looking a bit into other textbooks than mine and a few arXiv papers, I see that this my opinion is not standard, and I retract my argument. :) Kyle Barbour 20:55, 5 April 2008 (UTC)

## Error in The First 35 Twin Prime Pairs?

Hello -

The following statement under "The First 35 Twin Prime Pairs" heading seems wrong:

"Every third odd number greater than seven is divisible by 3"

This should probably be "Nine and every third odd number following is divisible by 3", or "Every third odd number greater than three is divisible by 3" (this last probably makes the most sense in context).

In the original, 13 is the third odd number greater than seven, followed in series by 19, 25, 31, 37, etc., none of which is evenly divisible by 3.

TablaRasa (talk) 18:41, 21 January 2010 (UTC)

Interesting. It would never have occurred to me to interpret "every third odd number greater than seven" as meaning "every third odd number starting with the third one above seven". To me the natural reading is that, among odd numbers above seven, multiples of 3 occur regularly, with the each one followed by another 3 steps further on in the sequence, not that this is true and these multiples of 3 start at the third one after 7. However, since at least one person has read it that way, someone else may do so too, so the original wording had better not be restored. However, it is clear that "greater than 7" is relevant: the point is that from 7 onwards 3 successive odd numbers cannot all be primes, so to rephrase it to say "greater than 3", as has been done, misses the point. I have rewritten it in a form which I hope unambiguously expresses the intended meaning, though to me the original wording seemed more straightforward. JamesBWatson (talk) 09:19, 26 January 2010 (UTC)
I intentionally wrote "greater than 3", because it is the 7 which misses the point, its mention is completely irrelevant and misleading. The wording with every third number above 7 wrongly suggested that (1) not every third number below 7 is divisible by 3, and (2) that 7 has something to do with the nonexistence of bigger "doubly twin" primes. Both are wrong. That there are no other twin primes belonging to two pairs has nothing to do with the largest of the three numbers being greater than 7, the real reason is that none of the three numbers equals 3. — Emil J. 11:19, 26 January 2010 (UTC)
Yes, I see what you mean. However, I think the original wording was meant to mean something like "from 7 upwards 3 successive odd numbers cannot be prime because every third one is a multiple of 3 and therefore composite", and it would not have occurred to me to read it as suggesting either of your points (1) and (2). Nevertheless, I agree that your current wording is clearer, so thank you. JamesBWatson (talk) 08:42, 28 January 2010 (UTC)

## Recursive Definition

The article starts with "Sometimes the term twin prime is used for a pair of twin primes". Isn't that a recursive definition? In other words, what is it trying to say? --92.107.32.139 (talk) 22:35, 2 May 2011 (UTC)

It's saying that the pair (11,13) is a twin prime, rather than saying that 11 is a twin prime by itself or that 13 is a twin prime by itself. —David Eppstein (talk) 22:59, 2 May 2011 (UTC)

## Merger proposal

The following discussion is closed. Please do not modify it. Subsequent comments should be made in a new section. A summary of the conclusions reached follows.
The result was merge into Twin prime. -- Toshio Yamaguchi (talk) 15:55, 13 May 2011 (UTC)

In accordance with a suggestion at User talk:PrimeHunter#Articles about classes of prime numbers, I suggest merging Isolated prime into this article. Please vote below so that consensus can be reached over wheather to proceed with the merger. Use the following format for your vote. Thanks. Toshio Yamaguchi (talk) 19:25, 7 May 2011 (UTC)

*'''Support''' - insert reason for supporting merger here ~~~~

*'''Oppose''' - insert reason for opposing merger here ~~~~

• Support - Very short and Isolated primes don't seem to be especially notable apert from Twin primes. Toshio Yamaguchi (talk) 19:36, 7 May 2011 (UTC)
• Support as original proposer. Isolated prime is just a rarely used term for a non-twin prime. Wikipedia is not a dictionary and the term fits better together with twin primes. PrimeHunter (talk) 19:43, 7 May 2011 (UTC)
• Support I'm not a number theorist and would defer to any who said there's something interesting to be said about isolated primes separate from a discussion of twin primes, but I can't really imagine what it might be. --Trovatore (talk) 23:42, 7 May 2011 (UTC)

The above discussion is closed. Please do not modify it. Subsequent comments should be made in a new section.

## Removed "citation needed"

I removed the citation-needed tag for this sentence: "The Hardy–Littlewood conjecture (after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture." Sufficient "citation" for this statement is the conjecture itself, which gives an approximation ${\displaystyle \pi _{2}(n)}$ of the number of twin prime pairs less than ${\displaystyle n}$. Since ${\displaystyle \pi _{2}(n)}$ becomes infinitely large as ${\displaystyle n}$ increases, clearly the twin primes conjecture is true if Hardy-Littlewood is. But H-L says more: it also gives the natural density of the twin primes. As I understand the term in this context, that is the definition of a generalization. 97.81.98.52 (talk) 15:47, 8 July 2012 (UTC)

— Preceding unsigned comment added by 77.238.65.171 (talk) 09:24, 3 April 2013 (UTC)

## Two more twin prime conjectures

I. If we have one twin prime pair (a1, a2) - the number X = 2 x (a1 + a2) always can be expressed as the sums of two prime numbers X = b1 + c2, that are members of twin prime pairs (b1, b2); (c1, c2).

Or another interpretation of this conjecture: There are always three primes p1, p2, p3 that are members of twin prime pairs (p1, p1+2); (p2, p2 + 2); (p3, p3 + 2) - for which the equation applies

p1 = [(p2 + p3 + 2)/ 4] – 1

P.s. Not excluding the possibility p1 = p2, or p1 = p3 (exam. 3, 3 , 11).

Example: 1. (11, 13); (17, 19); (29, 31); 48 = 2 x (11 + 13); 48 = 17 + 31 = 19 + 29; 11 = [(17 + 29 + 2)/4] - 1

2. (239, 241); (149, 151); (809, 811); 960 = 2 x (239 + 241); 960 = 149 + 811 = 151 + 809; 239 = [(149 + 809 + 2)/4] - 1

"Dan Zwillinger in 1979. He looked specifically at how even numbers between the values 2 --> 1,000,000 can be written as the sum of twin primes. This is a much denser superset of those tested in Enchev's conjecture..."

II. Between the squares of two consecutive odd numbers [2n+1]^2, and [2(n+1)+1]^2 there is always at least one pair of twin primes.

89.25.103.185 (talk) 12:15, 18 February 2017 (UTC)

## Zhang Yitang's result, is this correct?

"On April 17, 2013, Zhang Yitang announced a proof that for some integer N that is at most 70 million, there are infinitely many pairs of primes that differ by N."

I am reading Tao's note on this, and I think this statement misrepresent the result. According to my understanding, Zhang's proof managed to show that, there exist an infinite number of pair of prime such that their differences are no more than 70 million apart. However, what this statement said is that, there exist a number N<70 millions, such that once you fixed that number, there are infinitely many pair of primes which have differences being exactly N, which is a much stronger statement. — Preceding unsigned comment added by 143.44.68.136 (talkcontribs) 02:25, 7 June 2013‎

Actually the two ways of stating the result are equivalent. Given the formally weaker statement, there are infinitely many pairs of primes {p,q}, with q greater than p, such that qp is less than 70 million. So for each N less than 70 million, how many pairs are there with qp equal to N? For some N, the number of pairs may be finite, but if it were finite for all such N, you could add up the 70 million finite numbers and have a finite number of {p,q} in total, contradicting the assumption. --Trovatore (talk) 03:13, 7 June 2013 (UTC)
I stumbled over the wording too. Thank you for the explanation. — Preceding unsigned comment added by 129.233.1.162 (talk) 12:26, 30 July 2015 (UTC)

## Reader feedback: How big is the probability t...

95.222.167.105 posted this comment on July 11, 2012 (view all feedback).

How big is the probability to find a twin prime below a certain threshold

The answer to this is surely unknown - the Twin Prime Conjecture's truth or falsity would surely have a massive bearing on this. Smaug123 (talk) 15:58, 29 June 2013 (UTC)

Actually, the way the question is worded, the probability is 1 if the threshold is at least 5. Bubba73 You talkin' to me? 16:33, 29 June 2013 (UTC)
If we know two consecutive twin primes, can we estimate the third? Let us use k_n to mean the pair [6(k_n)-1, 6(k_n)+1]. Let us collect only twin primes and arrange them in ascending order so that we have a set of ordered pairs {(n, k_n); n=1, 2, 3, ...}. A plot of k_n as a function of n is interesting, for k_n would vary nearly linearly with n over a relatively wide range of n. If that being true, then k_(n+1)≈2k_(n)-k_(n-1). Let us look at three examples:

(a) Estimate of (9195287, 9195289) from (9194441, 9194443) and (9194837, 9194839). The estimate yields k_(n+1)≈2×1532473–1532407=1532539, an error of 0.00059%. (b) Estimate of (13915631, 13915633) from (13915019, 13915021) and (13915301, 13915303). The estimate yields k_(n+1)≈2×2319217-2319170=2319264, an error of 0.00034%. (c) Estimate of (18409199, 18409201) from (18408749, 18408751) and (18408989, 18408991). The estimate yields k_(n+1)≈2×3068165-3068125=3068205, an error of 0.00016%.174.91.76.100 (talk) 15:59, 5 August 2013 (UTC)

The relevant error measure is in the difference to the next twin prime pair. From 9194441 to 9194837 there is a difference of 396, so you expect there to also be 396 to the next. There is actually 450. That's an error of 13.6%. Your other examples have similar errors, and most examples would have larger errors than that. The expected growth rate is shown in Twin prime#First Hardy–Littlewood conjecture. It's not linear. And even if it had been linear, we wouldn't expect low error rates when comparing consecutive gaps between twin primes. "Random" fluctuations are too large for that. PrimeHunter (talk) 01:46, 6 August 2013 (UTC)
If you care to plot k_n as a function of n, you will see the trend is pretty linear. The estimated k_n is very close to the true k_n. That is what I have observed. Thanks for your comments. 174.91.76.100 (talk) 20:45, 6 August 2013 (UTC)
It only looks pretty linear if you plot a limited interval. Try oeis:A007508, or http://sweet.ua.pt/tos/primes.html#t2 if you want more data points. As almost anyone would have expected, twin primes become more rare for as far as they have been counted. PrimeHunter (talk) 21:37, 6 August 2013 (UTC)

## Chris De Corte proved that the number of twin primes depend on the number of primes below 2n

This can be found on page 21 of his paper "Derivation of a Prime verification formula to prove the related open problems". — Preceding unsigned comment added by Chrisdecorte (talkcontribs) 04:21, 18 July 2013 (UTC)

Self-published. I can't tell if it's drivel, but I've never seen a serious result first published as a slide show. arXiv is more credible than that. — Arthur Rubin (talk) 04:37, 18 July 2013 (UTC)

## Insufficient information in History section

The History section ought to state all the updates of what the upper bound for the smallest prime gap that occurs infinitely many times has reduced to since Yitang Zhang has found the first upper bound of 70,000,000 but this article doesn't mention a thing about the upper bound having been 12,006 before. Blackbombchu (talk) 17:27, 13 August 2013 (UTC)

What would be the point? The polymath project is a work in progress. It is neither reasonable nor in any way useful to list all the 70 or so updates they found in short succession, which are documented on the linked page for anyone who cares. The fact that the whole thing is mentioned here at all is already stretching WP:V quite far, as none of the results have been published in a reliable source.—Emil J. 19:12, 13 August 2013 (UTC)
The record is 4,680 now. This article should get updated in less than one day the next time the upper bound reduces. At the time the record was 70,000,000, I'm sure Yitang Zhang himself easisy could have lowered that bound a small bit himself if he even made the sligtest effort to do so based on the fact that 70,000,000 just so happens to be a digit followed by a string of zeros, so it may not be entirely correct to say the bound was 70,000,000 at that time. The record can be found in the sixth reference and the third reference is a dead link. Blackbombchu (talk) 21:03, 15 August 2013 (UTC)
Wikipedia is not a newspaper, and the world will not end tomorrow. There is no hurry to have everything updated at any given minute, and you are not in position to issue any ultimata. Having said that, you are not reading the table correctly, the question mark means that 4680 is “unconfirmed or conditional”, so the number in the article is correct at the moment. I fail to see any relevance in your speculation on Yitang Zhang.
Reference #3 is a link to the paper at the Annals of Mathematics, chances are you can’t see it because you have no subscription. However, linking directly to the pdf is not a good idea, I’ll change that to the abstract page.—Emil J. 12:00, 16 August 2013 (UTC)

## A165959

A quote in OEIS for A165959 "If this sequence has an infinite number of terms in which a(n) = 3, then the twin prime conjecture can be proved." This is related to Ramanujan_prime. --John W. Nicholson (talk) 02:47, 13 November 2013 (UTC)

## More current

"Unconditionally, our bound on {H_1} is still {H_1 \leq 270}."

http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271926%20264

The Twin Prime Conjecture proof is being published 3rd quarter this year. It just made it past peer review. — Preceding unsigned comment added by 72.48.254.167 (talk) 07:39, 20 April 2016 (UTC)

An edit war is in danger of breaking out here! The issue at stake is the sentence in the History section: "On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by at most N.[1][2]" and whether it should say "...differ by at most N" or just "differ by N".

From the Nature paper referenced at [1] and [2]: "The new result, from Yitang Zhang ..., finds that there are infinitely many pairs of primes that are less than 70 million units apart". To me this seems to mean, "for some N less than 70 million, there are infinitely many pairs of primes that differ by less than N". Any further opinions?: Noyster (talk), 14:14, 7 April 2014 (UTC)

There are two main ways to formulate the same result:
1) For some (i.e. for at least one) N less than 70 million, there are infinitely many pairs of primes that differ by N
2) There are infinitely many pairs of primes that differ by less than 70 million
2) trivially implies 1) since there are only finitely many possible differences below 70 million, so if they have infinitely many occurrences in total then at least one of them must have infinitely many occurrences.
Some editors want this version:
"For some (i.e. for at least one) N less than 70 million, there are infinitely many pairs of primes that differ by at most N"
That's technically also correct but it's a silly mix of 1) and 2) to first say "N less than 70 million" and then "at most N". If there are infinitely many that differ by at most N, then there are obviously also infinitely many (the same) that differ by at most M, for any M ≥ N (for example for M = 70 million if N was less than 70 million). PrimeHunter (talk) 14:38, 7 April 2014 (UTC)

## Just a question of elementary number theory the twin prime conjecture

Expand to see the material in question
The following discussion has been closed. Please do not modify it.

### There are infinitely many primes P and P + 2 is a prime number

The twin prime formula Using the criterion of prime numbers, we can get the following conclusions: "if the natural numbers ${\displaystyle q}$ and ${\displaystyle q+2}$ cannot be any less than${\displaystyle {\sqrt {q+2}}}$ prime divisibility, then ${\displaystyle q}$ and ${\displaystyle q+2}$ is prime.。 This is because a natural number${\displaystyle n}$ is prime if and only if it can't be any less than other prime divisibility on${\displaystyle {\sqrt {n}}}$ . Using mathematical language expresses the conclusions above, is: There is a group of natural number${\displaystyle b_{1},b_{2}\cdot ,b_{k}}$ making

${\displaystyle q=p_{1}m_{1}+b_{1}=p_{2}m_{2}+b_{2}=\dots =p_{k}m_{k}+b_{k}\qquad \qquad \qquad \cdots \qquad (1)}$

The${\displaystyle p_{1},p_{2},\dots ,p_{k}}$ from small to large order during the first k prime numbers: 2, 3, 5,.... And meet。

${\displaystyle \forall 1\leq i\leq k,\ \ 0

Natural number ${\displaystyle q}$ so obtained if meet the ${\displaystyle q${\displaystyle q}$and${\displaystyle q+2}$

are a pair of twin primes. We can think of (1) equivalence of content type conversion become congruence equations。

${\displaystyle q\equiv b_{1}{\pmod {p_{1}}},q\equiv b_{2}{\pmod {p_{2}}},\dots ,q\equiv b_{k}{\pmod {p_{k}}}\qquad \qquad \qquad \cdots \qquad (2)}$

Because of (2).${\displaystyle p_{1}}$,${\displaystyle p_{2}}$,...,${\displaystyle p_{k}}$ are all prime numbers, so two two coprime,based on the Chinese Remainder Theorem (China remainder theorem), for a given${\displaystyle b_{1},b_{2}\cdot ,b_{k}}$ , (2) is only a less than positive integer${\displaystyle p_{1}p_{2}\cdots p_{k}}$ solution。.

### Example

For example, k=1,${\displaystyle q=2m_{1}+1}$，solution${\displaystyle q=3,5}$ 。 Since${\displaystyle 5<3^{2}-2}$

so that ${\displaystyle 3}$ and${\displaystyle 3+2}$ ；${\displaystyle 5}$ and${\displaystyle 5+2}$  are the twin prime. Thus the interval${\displaystyle (3,3^{2})}$  in the
whole of prime twins.。


For instance

k=2, equation of ${\displaystyle q=2m_{1}+1=3m_{2}+2}$，  solution${\displaystyle q=5,11,17}$。Since${\displaystyle 17<5^{2}-2}$，
so the${\displaystyle 11}$  with${\displaystyle 11+2}$  ；${\displaystyle 17}$ and ${\displaystyle 17+2}$ are the twin primes.


Because this isall possible ${\displaystyle b_{1},b_{2}\cdot ,b_{k}}$ value, so the interval ${\displaystyle (5,5^{2})}$ are of prime twins.。

k=3 ${\displaystyle 5m_{3}+1}$ ${\displaystyle 5m_{3}+2}$ ${\displaystyle 5m_{3}+4}$
${\displaystyle q=2m_{1}+1=3m_{2}+2}$= 11,41 17 29

Because this is all possible${\displaystyle b_{1},b_{2}\cdot ,b_{k}}$ value, so the interva${\displaystyle (7,7^{2})}$are of prime twins。

k=4 ${\displaystyle 7m_{4}+1}$ ${\displaystyle 7m_{4}+2}$ ${\displaystyle 7m_{4}+3}$ ${\displaystyle 7m_{4}+4}$ ${\displaystyle 7m_{4}+6}$
${\displaystyle q=2m_{1}+1=3m_{2}+2=5m_{3}+1}$= 71 191 101 11 41
${\displaystyle q=2m_{1}+1=3m_{2}+2=5m_{3}+2}$= 197 107 17 137 167
${\displaystyle q=2m_{1}+1=3m_{2}+2=5m_{3}+4}$= 29 149 59 179 209

Because this is all possible${\displaystyle b_{1},b_{2}\cdot ,b_{k}}$ value, so the interva${\displaystyle (11,11^{2})}$are of prime twins。

8 less than the 121-2 of the solution。

Copy this to be a not leak to find all the prime twins within an arbitrary largenumber of. For all the possible${\displaystyle b_{1}}$},${\displaystyle b_{2}}$ ... ,${\displaystyle b_{k}}$ value, (1) and (2) in the（${\displaystyle p_{1}}$）（${\displaystyle p_{2}}$）(${\displaystyle p_{3}}$)...（${\displaystyle p_{k}}$）range,， with（${\displaystyle p_{1}-1}$）（${\displaystyle p_{2}-2}$）(${\displaystyle p_{3}-2}$)...（${\displaystyle p_{k}-2}$）.....（3）solution。

### Conclusion

The twin prime conjecture is in the K value arbitrarily large (1) and (2) type areless than${\displaystyle p_{k+1}^{2}-2}$ solution。. The problem has been transferred into the elementary number range.。

Is this material published in any independent reliable source? If not, we cannot use it. This page is not a forum to discuss unpublished or partially completed work in the subject: there are other plsces for that. Deltahedron (talk) 10:23, 6 September 2014 (UTC)

I give up fixing the < math > tags. I don't think the proof is correct, anyway, and, even if it were, as Deltahedron points out, we couldn't use it until published. — Arthur Rubin (talk) 02:47, 9 September 2014 (UTC)

## Missing term in list in the Other theorems ... section

In the paragraph, beginning "By assuming the Elliott–Halberstam conjecture ...", shouldn't there be a term n+14 at the obvious place in the list? --Stfg (talk) 19:06, 18 February 2017 (UTC)

No, the referenced paper [1] is about an admissible 7-tuple (n, n + 2, n + 6, n + 8, n + 12, n + 18, n + 20). All seven can be prime, e.g. for n = 11. If (n, n + 2, n + 6, n + 8) are all primes above 5 (a prime quadruplet) then 5 must divide n + 4 and n + 14. By the way (using a poor excuse to brag), I, Jens Kruse Andersen, have broken the record for the largest known prime 7-tuplet several times, but don't have the current 402-digit record: http://anthony.d.forbes.googlepages.com/ktuplets.htm#largest7. PrimeHunter (talk) 19:48, 18 February 2017 (UTC)
Oh yes, silly me. Thanks, and kudos for all those past records :) --Stfg (talk) 21:56, 18 February 2017 (UTC)