Brun's Constant

The number obtained by adding the reciprocals of the Twin Primes,

$$B \equiv \left({{\textstyle{1\over 3}}+{\textstyle{1\over 5}... ...\textstyle{1\over 17}}+{\textstyle{1\over 19}}}\right)+\cdots,$$ (1)

By Brun's Theorem, the constant converges to a definite number as $p\to\infty$. Any finite sum underestimates $B$. Shanks and Wrench (1974) used all the Twin Primes among the first 2 million numbers. Brent (1976) calculated all Twin Primes up to 100 billion and obtained (Ribenboim 1989, p. 146)

$$B\approx 1.90216054,$$ (2)

assuming the truth of the first Hardy-Littlewood Conjecture. Using Twin Primes up to $10^{14}$, Nicely (1996) obtained

$$B\approx 1.9021605778\pm 2.1\times 10^{-9}$$ (3)

(Cipra 1995, 1996), in the process discovering a bug in Intel's ${}^{\scriptstyle\circledRsymbol}$ Pentium${}^{\rm TM}$ microprocessor. The value given by Le Lionnais (1983) is incorrect.

See also Twin Primes, Twin Prime Conjecture, Twin Primes Constant

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 64, 1987.

Brent, R. P. ``Tables Concerning Irregularities in the Distribution of Primes and Twin Primes Up to $10^{11}$.'' Math. Comput. 30, 379, 1976.

Cipra, B. ``How Number Theory Got the Best of the Pentium Chip.'' Science 267, 175, 1995.

Cipra, B. ``Divide and Conquer.'' What's Happening in the Mathematical Sciences, 1995-1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 38-47, 1996.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/brun/brun.html

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 41, 1983.

Nicely, T. ``Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant.'' Virginia J. Sci. 46, 195-204, 1996. http://lasi.lynchburg.edu/Nicely_T/public/twins/twins.htm.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, 1989.

Shanks, D. and Wrench, J. W. ``Brun's Constant.'' Math. Comput. 28, 293-299, 1974.

Wolf, M. ``Generalized Brun's Constants.'' http://www.ift.uni.wroc.pl/~mwolf/.