1,4

Goldbach conjectured in 1742 that this sequence never vanishes. This is still unproved.

Number of different primes occurring when 2n is expressed as p1+q1 = ... = pk+qk where pk,qk are odd primes with pk <= qk. For example when n=5: 10 = 3+7 = 5+5, we can see 3 different primes so a(5) = 3. - Naohiro Nomoto (n_nomoto(AT)yabumi.com), Feb 24 2002

Comments from Tomas Oliveira e Silva to Number Theory List, Feb 05 2005: With the help of Siegfied "Zig" Herzog of PSU, I was able to verify the Goldbach conjecture up to 2e17. Let 2n=p+q, with p and q prime be a Goldbach partition of 2n. In a minimal Goldbach partition p is as small as possible. The largest p of a minimal Goldbach partition found was 8443 and is need for 2n=121005022304007026. Furthermore, the largest prime gap found was 1220-1; it occurs after the prime 80873624627234849.

Comments from Tomas Oliveira e Silva to Number Theory List, Apr 26 2007: With the help of Siegfried "Zig" Herzog, the NCSA, and others, I have just finished the verification of the Goldbach conjecture up to 1e18. This took about 320 years of CPU time, including a double-check of the results up to 1e17. As expected, no counter-example to the conjecture was found. As side results, the number of twin primes up to 1e18 was also computed, as was the number of primes in each of the residue classes modulo 120. Also, the number of occurrences of each (observed) prime gap was also recorded.

For n>2 we have a(n)=2A002375(n)-1 if n is prime and a(n)=2A002375(n) if n is composite. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 14 2004

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.

R. K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.

G. H. Hardy and J. E. Littlewood, Some problems of `partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 2.8 (for Goldbach conjecture).

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 79, 80.

N. Pipping, Neue Tafeln fuer das Goldbarsche Gesetz nebst Berichtungen zu den Haussnerschen Tafeln, Finska Vetenskaps-Societeten, Comment. Physico Math.. 4 (No. 4, 1927), 27 pp.

J. Richstein, Verifying the Goldbach conjecture up to 4*10^14, Mathematics of Computation, Vol. 70, No. 236, pp. 1745-1749, 2000.

Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Mathematics of Computation, Vol. 61, No. 204, pp. 931-934, October 1993.

M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.

T. D. Noe, Table of n, a(n) for n = 1..10000

T. Oliveira e Silva, Goldbach conjecture verification

T. Oliveira e Silva, Title?

T. Oliveira e Silva, Title?

Eric Weisstein's World of Mathematics, Link to a section of The World Of Mathematics

Index entries for sequences related to Goldbach conjecture

A. Zaccagnini, Goldbach Variations: problems with prime numbers.

2? No! 4? No! 6 = 3+3 so a(3)=1. 8 = 3+5 = 5+3 so a(4) = 2. 10 = 5+5 = 3+7 = 7+3 so a(5) = 3. etc.

a(5)=3 because 10=3+7=5+5=7+3.

a:=proc(n) local c, k; c:=0: for k from 1 to n do if isprime(2*k+1)=true and isprime(2*n-2*k-1)=true then c:=c+1 else c:=c fi od end: seq(a(n), n=1..82); (Deutsch)

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n-1, i++ If[OddQ[i]&&PrimeQ[i]&&PrimeQ[2n-i], cnt++ ]]; AppendTo[lst, cnt]]; lst

Essentially identical to A035026.

Cf. A002375 (unordered sums), A002374, A014092, A035026, A059998, A001031, A002373, A045917, A006307.

Adjacent sequences: A002369 A002370 A002371 this_sequence A002373 A002374 A002375

Sequence in context: A054237 A129600 A081388 this_sequence A035026 A070770 A071487

nonn,nice,easy

njas

More terms from Larry Reeves (larryr(AT)acm.org), Jun 13 2002