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 × 10
11. 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 10
14. 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 [10
5i
, 10
5i
+10
8], for
i=3, 4,..., 20 and [10
10i
, 10
10i
+ 10
9], 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 11P32 - 11Y99
1991 Computing Reviews Classification System F.2.1
Keywords and Phrases Goldbach conjecture - sum of primes - primality test - vector computer - Cray C916 - cluster of workstations
To appear in the Proceedings of the Algorithmic Number Theory Symposium III (Reed College, Portland, Oregon, USA, June 21–25,
1998).