Goldbach's conjecture

Goldbach wrote a letter to Euler dated June 7, 1742 suggesting (roughly) that every even integer is the sum of two integers p and q where each of p and q are either one or odd primes.  Now we often word this as follows:

Goldbach's conjecture: Every even integer n greater than two is the sum of two primes.

This is easily seen to imply

Every integer n greater than five is the sum of three primes.

There is little doubt that both results are true, as Euler replied to Goldbach:

That every even number is a sum of two primes, I consider an entirely certain theorem in spite of that I am not able to demonstrate it.

Progress has been made on this problem, but slowly--it may be quite awhile before the work is complete.  For example, it has been proven that every even integer is the sum of at most six primes (Goldbach suggests two) and in 1966 Chen proved every sufficiently large even integer is the sum of a prime plus a number with no more than two prime factors (a P₂).

Vinogradov in 1937 showed that every sufficiently large odd integer can be written as the sum of at most three primes, and so every sufficiently large integer is the sum of at most four primes.  One result of Vinogradov's work is that we know Goldbach's theorem holds for almost all even integers.

Various heuristic estimates are avaliable for the number of solutions there should be to 2n = p + q (with p, q prime). And these of course grew in size with n.

Recently, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have verified this up to 10¹⁴ with the help of a Cray C90 and various workstations.  If we will accept the Riemann Hypothesis, then this is enough to prove the odd Goldbach conjecture: Every odd integer greater than five is the sum of three primes.

Descartes also was aware of the two prime version of Goldbach's conjecture before Goldbach was.  So is it misnamed?  Erdös said that it "is better that the conjecture be named after Goldbach because, mathematically speaking, Descartes was infinitely rich and Goldbach was very poor."

When verifying the Goldbach conjecture for n we quickly see that it is very easy to find many primes which add to n.  In 1923 Hardy and Littlewood took the first major step toward the proof of the Goldbach conjectures using their circle method [HL23].  Among other things, they conjectured that the number of ways of writing n as the sum of two primes, G(n), is asymptotic to twice the twin prime constant times n/(log n)² times the product of (p-1)/(p-2) taken over the prime divisors p of n.

That is a lot of solutions for a large n so there will be a solution with one of the prime quite small.  For example, when checking all n up to 100,000,000,000,000, Richstein [Richstein2000] found the smaller prime is never larger than 5569 used in the following partition:

389965026819938 = 5569 + 389965026814369.

It was conjectured [GRL89] that the smallest prime in a partition of n might be bounded by a constant times (log n)² log log n.  Richstein's work shows that the constant 1.603 (from n=6) sufficies for n < 4^.10¹⁴.

See Also: OddGoldbachConjecture

Related pages (outside of this work)

• Prime conjectures
• Mixed Sums of Primes and Other Terms by Zhi-Wei Sun

References:

DRS98

J. M. Deshouillers, H. J. J. te Riele and Y. Saouter, New experimental results concerning the Goldbach conjecture.  In "Proc. 3rd Int. Symp. on Algorithmic Number Theory," Lecture Notes in Computer Science Vol, 1423, 1998.  pp. 204--215, MR 2000j:11143

FR89

Fliegel, H. F. and Robertson, D. S., "Goldbach's comet: the numbers related to Goldbach's conjecture," J. Recreational Math., 21:1 (1989) 1--7.

GRL89

A. Granville, H. J. J. te Riele and J. van de Lune,"Checking the Goldbach conjecture on a vector computer" in Number theory and its applications.  R. A. Mollin editor, Kluwer, 1989.  Dordrect, pp. 423--433,

HL23

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 Math., 44 (1923) 1-70.  Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon Press, Oxford, 1966.

Ramare95

O. Ramaré, "On Schnirelmann's constant," Ann. Sc. Norm. Super. Pisa, 22:4 (1995) 645-706.  MR 97a:11167

Ribenboim95

P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]

Richstein2000

J. Richstein, "Verifying the goldbach conjecture up to 4· 10¹⁴," Math. Comp., 70:236 (2001) 1745--1749.  MR 2002c:11131 (Abstract available)

Sinisalo93

M. Sinisalo, "Checking the Goldbach conjecture up to 4· 10¹¹," Math. Comp., 61:204 (1993) 931--934.  MR 94a:11157

Wang84

Y. Wang editor, Goldbach conjecture, Series in Pure Mathematics Vol, 4, World Scientific, 1984.  Singapore, pp. xi+311, ISBN 9971-966-08-5; 9971-966-09-3. MR 87h:11102 (Annotation available)