Goldbach Partition

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

(1)

The Goldbach conjecture is then equivalent to the statement that  or, equivalently, that , for every even integer .

A plot of , sometimes known as Goldbach's comet, for  up to 2000 is illustrated above.

The following table summarizes the values of several variants of  for , 4, ....

partition type	OEIS	values
1 or prime	A001031	1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, ...
prime	A045917	0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, ...
odd prime	A002375	0, 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).

REFERENCES:

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. https://arxiv.org/abs/nlin.CD/0601024.

Oliveira e Silva, T. "Goldbach Conjecture Verification." https://www.ieeta.pt/~tos/goldbach.html.

Sinisalo, M. K. "Checking the Goldbach Conjecture up to ." 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."