Talk:Brun's constant

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Axel: Brun's constant is denoted as B but Nicely (I don't know why) uses B₂ so I don't see the meaning of change. Please see other references too.
XJamRastafireRootsRockReggaeSecurityInvestigator [2002.02.27] 3 Wednesday somewhere outa space.

Well, you used both B and B₂ in your article, and I decided that one name for the number is enough. If you prefer B, please change it here and also in mathematical constants. AxelBoldt

No need. Let it stay B₂ and let future shows up the decision. Perhaps it would come out that we should name this constant Brun - Hauschaeckell's B_3z constant. Who knows --XJam [2002.02.27] 3 Wednesday (2nd ed.)

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Axel, is this better?

1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers p and q which differ by two) B₂(p,q):

B₂(p,q) = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + (1/29 + 1/31) + (1/41 + 1/43) + (1/59 + 1/61) + ...

converges to a finite constant now called Brun's constant for twin primes and thus usually denoted by B₂ and defined as:

B₂ = lim_p,q→∞ B₂(p,q).

I think we should 'somehow' distinguish between the sum B₂(p,q) and the Brun's constant B₂.

XJam [2002.04.02] 2 Tuesday (0)

No, your notation is unclear: you write B₂(p,q) for a number that doesn't depend on p and q! Your definition of B₂(p,q) above is exactly Brun's constant; the limit is already built in because of the ... in the formula. It is an infinite series.

I think the definition in the article is clear right now. AxelBoldt, Tuesday, April 2, 2002

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Ralf: The link to 'Pascal Sebah' leads to a 19th century photographer.

B_n generalizations

This constant should not be confused with the Brun's constant for cousin primes,
prime pairs of the form (p, p + 4), which is also written as B4.
Wolf derived an estimate for the Brun-type sums Bn of 4/n.
This gives the estimate for Bn of 2, about 5% higher than the true value.

Firstly, it's not entirely clear at first glance whether the second sentence refers to the "cousin" type of B_n that was just defined, or if the two sentences (which were added at different times) are adjacent by coincidence and the second one is referring to the "quadruplet" type of B_n. (The answer seems to be that it's the "cousin" type -- neighboring primes with a difference of exactly n.) Secondly, I just looked at the abstract of Wolf's paper, and the formula is not 4/n, but rather something more complicated with a 4 in the numerator and an n in the denominator. Thirdly, Wolf's result only applies to n >= 6, so even if the 4/n formula is somehow correct, applying it to B₂ strikes me as misleading or pointless. Fourthly, I'm a little bothered by calling 1.902 the "true value", when it's stated earlier that no upper bound for the value has been proved. I'm inclined to delete everything after the first sentence, but I'll leave it alone for now and see if anyone else agrees or disagrees. 76.202.61.191 (talk) 06:03, 19 May 2008 (UTC)

Can anyone please explain for me, briefly, why an irrational Brun's constant implies infinitude of twin primes? Thanks.209.167.89.139 (talk) 13:48, 10 August 2009 (UTC)

If there were a finite number of twin primes, then Brun's constant would be a finite sum of rational numbers, so it would be rational itself. So, if we find out that Brun's constant is irrational, that would mean that there isn't a finite number of twin primes. --Sfztang (talk) 05:17, 16 October 2009 (UTC)

Bound on B2

"While 1.9 < B2 is shown, no real number N is known such that B2 < N." What about N=2? Am I missing something? – Sigbhu —(talk • contribs) 15:10, 15 February 2010 (UTC)

In [1] I have written about the estimate 1.902160583104:

"It is based on extrapolation from the sum 1.830484424658... for the twin primes below 10¹⁶. While 1.83 < B₂ is shown, no real number N is known such that B₂ < N.^{[Citation needed]}"

It is my understanding it has only been proved that B₂ is finite and at least 1.830484424658... which is the sum for twin primes below 10¹⁶. The rest is only a heuristic guess based on the expected frequency of twin primes above 10¹⁶. The twin primes below 10¹⁶ follow the expectation well and it appears likely that those above will also do it, but large twin primes could theoretically be common enough to make B₂ much larger than 2, or rare enough to make it below 1.9. I have therefore removed the claim that the best estimate is 1.902160583104. Maybe it is true that it is the best estimate but it has not been proved to be the best (assuming "best" means closest to the real value). PrimeHunter (talk) 02:14, 16 February 2010 (UTC)

Crandall & Pomerance

Someone should get and read their book. There's no proof in it.Eganfan (talk) 12:41, 3 July 2010 (UTC)

So tag it as {{verification failed}}, then. Real bounds would be notable. — Arthur Rubin (talk) 18:14, 3 July 2010 (UTC)

No real bounds are known. Eganfan (talk) 21:22, 3 July 2010 (UTC)

I have reverted your removal with edit summary "Crandall & Pomerance is a reliable source and verifiably says B_2 < 2.347, accepted by others, Wikipedia does not require proofs in reliable sources". See also for example page 1 of http://www.math.uiuc.edu/~pppollac/polyapps.pdf which says: "The upper bound is due to Crandall and Pomerance ([5, pp. 16-17], see also [19, Chapter 3]), who bound the sum of the twin prime pairs past 10¹⁶ using an explicit upper estimate of Riesel and Vaughan [29] for the number of twin prime pairs." PrimeHunter (talk) 23:05, 3 July 2010 (UTC)

I don't see your point. All I'm saying is there is no proof of B_2 < 2.347. Eganfan (talk) 08:37, 4 July 2010 (UTC)

I have reverted you again. We don't need a proof in the cited source when the source is reliable. It is cited as a source for the claim "B_2 < 2.347", not for a mathematical proof of that claim. The citation satisfies Wikipedia:Verifiability. Wikipedia uses the same citation rules for all the millions of articles. Few of them are about subjects where a mathematical proof of correctness is possible. We report what reliable sources say. If you can find another reliable source contradicting the respected Crandall & Pomerance then we can discuss it, but otherwise you should stop edit warring. PrimeHunter (talk) 21:21, 4 July 2010 (UTC)

What are you talking about? No one ever proved "B_2 < 2.347". Are you religious person or what?Eganfan (talk) 21:49, 4 July 2010 (UTC)

Clearly the reference is not ideal. It would be nice if it gave a proof or had a reference to the original source. Nevertheless it is a serious scientific source and I see no reason not to trust them. Especially since experts in this comment thread seemed convinced that one should be able to get some upper bound using Brun's method. I am surprised by Eganfan's confident insistence to the contrary. I tried asking on Mathoverflow but people there were not aware of any better references. MathHisSci (talk) 09:08, 10 October 2010 (UTC)