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There are remarkable patterns of density in the graph http://upload.wikimedia.org/wikipedia/commons/7/7c/Goldbach-1000000.png plotting the number of representations of even numbers up to a million as the sum of two primes. Has anyone measured these patterns? The referenced graph was uploaded in 2006. Given the enormous data bases of primes, it would seem to be a matter of simple checking of sums to generate the graph, with a complexity that grows not much more than linearly with the size of the even number. Thus, with the growth of computer power, some advance on a million might have been achieved over the past few years. Has this been done?

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2 Answers 2

That link reveals that a certain heuristic but extremely plausible assymptotic formula is highly accurate at least up to $10^{10}$. That information tells us what a graph over a larger range would look like. So one knows certain patterns are there even if the graph is not drawn. (Maybe someone has drawn it, I don't know).

Roughly, the expected number of pairs is about $0.66\frac{n}{{\log}^2(n)}$ for a large number which is twice a prime. Multiply that by $\prod_{p|n}\frac{p-1}{p-2}$ to get the estimate for any even n (the product over odd prime divisors of $n$).

This explains these patterns (do you see others?): There should be a lower half very roughly hitting at $10^6$ from 3460 to 6700 and then an upper half from 6920 to 13400 about twice the lower part. The lower part is for 2 or 4 mod 6 and the upper for mutiples of 6. Numbers which are or are not multiples of 5 and/or 7 should create 4 bands in each half (I can't see much beyond that). Those patterns (including with more primes taken into account)continue very faithfully as far as calculated.

added information

The name "Goldbach's Comet" is probably not useful in finding extended computations. "Goldbach partitions" might do better. As far as I can tell the state of the art is

MR1850627 (2002g:11142) Richstein, Jörg . Computing the number of Goldbach partitions up to $5\cdot 10^8$. Algorithmic number theory (Leiden, 2000), 475--490, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.

That paper does include a graph for the range $500500000 \le n \le 500660160$. The author no doubt has the data and could generate other graphs, but may not see any reason to. There are a number of observations in that paper about patterns. You might be better served by looking at color coded plots over short ranges (say according to congruence class mod 30 or 210). for example a plot over a modest range colored mod 6 shows that the top is all of the $0 \mod 6$ and nothing else (of course) BUT it also shows that the very bottom boundry is heavily in favor of $2 \mod 6$ leading me to pose this question.

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Thank you for the search term "Goldbach comet", given in your edit to Gerry Myerson's answer, which does indeed lead to many sites via Google. However, I have not been able to find among them plots of the comet even as far as the original reference's $10^6$: they are mostly relatively old sites where the plot doesn't reach $10^5$. –  John Bentin Feb 2 '11 at 21:27
I just fixed the link. What are you looking for in a plot with a larger range? It will look "the same" only more. –  Aaron Meyerowitz Feb 2 '11 at 21:49
We would expect it to look similar over a larger range, from the Hardy--Littlewood formula. But, as far as I know, that formula is still only a conjecture. Moreover, the formula was proposed only as an asymptotically correct description, although it may be more precise than just that. The definition of the bands in the comet improves greatly as the graph is extended. By eye-balling pictures with very clear, regular, and measurable patterns, one can get ideas. I don't have any prior view on what I would expect to see in an extended graph. –  John Bentin Feb 3 '11 at 12:10
Ok I expanded the answer –  Aaron Meyerowitz Feb 3 '11 at 21:20

Yes. See http://www.ieeta.pt/~tos/goldbach.html

EDIT: The graph is sometimes referred to as the Goldbach Comet, and you'll find numerous links by typing that phrase into your favorite search engine. http://en.wikipedia.org/wiki/Goldbach's_comet for example covers some of the same ground that Aaron does in his answer.

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Thanks for the link. Although the site doesn't consider patterns in the referenced graph, it indicates (without reference) that enough data have been collected to extend the graph to an abscissa of $10^{10}$ ─ and perhaps more. –  John Bentin Feb 1 '11 at 16:13

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