The Goldbach conjecture states that every even integer ≥ 4 can be written as a sum of two prime numbers. It is known to be true up to 4 × 1011. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R10000 CPUs are described, which extend this bound to 1014. Two consequences are that (1) under the assumption of the Generalized Riemann hypothesis, every odd number ≥7 can be written as a sum of three prime numbers, and (2) under the assumption of the Riemann hypothesis, every even positive integer can be written as a sum of at most four prime numbers. In addition, we have verified the Goldbach conjecture for all the even numbers in the intervals [105i, 105i +108], for i=3, 4,..., 20 and [1010i, 1010i + 109], for i=20,21,..., 30.
A heuristic model is given which predicts the average number of steps needed to verify the Goldbach conjecture on a given interval. Our experimental results are in good agreement with this prediction. This adds to the evidence of the truth of the Goldbach conjecture.
1991 Mathematics Subject Classification
1991 Computing Reviews Classification System
Keywords and Phrases
Goldbach conjecture sum of primes primality test vector computer Cray C916 cluster of workstations
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J-M. Deshouillers, G. Effinger, H. te Riele and D. Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electronic Research Announcements of the AMS 3 (1997), pp. 99–104 (September 17, 1997); http://www.ams.org/journals/era/home-1997.html.MATHCrossRefGoogle Scholar
A. Granville, J. van de Lune and H.J.J. te Riele, Checking the Goldbach conjecture on a vector computer, Number Theory and Applications (R.A. Mollin, ed.), Kluwer, Dordrecht, 1989, pp. 423–433.Google Scholar
G.H. Hardy and L.E. Littlewood, Some problems of ‘Partitio Numerorum'; III: On the expression of a number as a sum of primes, Acta Math. 44 (1922/1923), pp. 1–70.CrossRefGoogle Scholar