Goldbach Partition
المؤلف:
Clawson, C
المصدر:
Mathematical Mysteries: The Beauty and Magic of Numbers. New York: Plenum Press
الجزء والصفحة:
...
16-1-2021
2735
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
![R(2n)=<span style=]() {2r(2n)-1 for n prime; 2r(2n) for n composite. " src="https://mathworld.wolfram.com/images/equations/GoldbachPartition/NumberedEquation1.gif" style="height:41px; width:239px" /> |
(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."
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