Goldbach Partition

A pair of primes (p,q) that sum to an even integer 2n=p+q are known as a Goldbach partition (Oliveira e Silva). Letting r(2n) denote the number of Goldbach partitions of 2n without regard to order, then the number of ways of writing 2n as a sum of two prime numbers taking the order of the two primes into account is

 R(2n)={2r(2n)-1   for n prime; 2r(2n)   for n composite.

The Goldbach conjecture is then equivalent to the statement that r(n)>0 or, equivalently, that R(n)>0, for every even integer n>1.


A plot of r(2n), sometimes known as Goldbach's comet, for n up to 2000 is illustrated above.

The following table summarizes the values of several variants of r(n) for n=2, 4, ....

partition typeOEISvalues
p,q 1 or primeA0010311, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, ...
p,q primeA0459170, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ...
p,q odd primeA0023750, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ...

Various fractal properties have been observed in Goldbach's partition (Liang et al. 2006).

See also

Goldbach Conjecture, Goldbach Number

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Clawson, C. Mathematical Mysteries: The Beauty and Magic of Numbers. New York: Plenum Press, p. 241, 1996.Doxiadis, A. Uncle Petros and Goldbach's Conjecture. Faber & Faber, 2001.Grave, D. A. Traktat z Algebrichnogo Analizu, Vol. 2. Kiev, Ukraine: Vidavnitstvo Akademiia Nauk, p. 19, 1938.Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974.Lehmer, D. H. Guide to Tables in the Theory of Numbers. Bulletin No. 105. Washington, DC: National Research Council, p. 80, 1941.Liang, W.; Yan, H.; and Zhi-cheng, D. "Fractal in the Statistics of Goldbach Partition." 12 Jan 2006. e Silva, T. "Goldbach Conjecture Verification.", M. K. "Checking the Goldbach Conjecture up to 4·10^(11)." Math. Comput. 61, 931-934, 1993.Sloane, N. J. A. Sequences A001031/M0213, A002375/M0104, and A045917 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Goldbach Partition

Cite this as:

Weisstein, Eric W. "Goldbach Partition." From MathWorld--A Wolfram Web Resource.

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