New experimental results concerning the Goldbach conjecture

The Goldbach conjecture states that every even integer $ge4$ can be written as a sum of two prime numbers. It is known to be true up to $4times 10^{11$. In this paper, new experiments on a Cray C916 supercomputer and on an SGI compute server with 18 R8000 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 $ge7$ 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 $i=3,4,dots,20$ and $[10^{10i, 10^{10i+10^9]$, for $i=20,21,dots,30$. 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.