Talk:Brun's constant

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)