Abstract of: New experimental results concerning the Goldbach conjecture

MAS-R9804
J-M. Deshouillers ; H.J.J. te Riele ; Y. Saouter ;
1998, MAS-R9804, ISSN 1386-3703
PDF-file compressed PostScript ( 106 KB ) ( 12 pages )
 
Abstract
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 R8000 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.

 
CWI Group(s):
MAS2 (Scientific Computing and Control Theory)
 
CWI Project(s):
Computational number theory
 
Keywords:
Goldbach conjecture sum of primes primality test vector computer Cray C916 cluster of workstations