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International Algorithmic Number Theory Symposium

ANTS 1998: Algorithmic Number Theory pp 204–215Cite as

New experimental results concerning the Goldbach conjecture

Part of the Lecture Notes in Computer Science book series (LNCS,volume 1423)

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 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

  • 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).

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Authors and Affiliations

  1. Mathématiques Stochastiques, Université Victor Segalen Bordeaux 2, F-33076, Bordeaux Cedex, France

    J. -M. Deshouillers

  2. Centre for Mathematics and Computer Science, CWI, Kruislaan 413, 1098, SJ Amsterdam, The Netherlands

    H. J. J. te Riele

  3. Institut de Recherche en Informatique de Toulouse, 118 route de Narbonne, F-31062, Toulouse Cedex, France

    Y. Saouter

Authors
  1. J. -M. Deshouillers
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  2. H. J. J. te Riele
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  3. Y. Saouter
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    © 1998 Springer-Verlag Berlin Heidelberg

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    Deshouillers, J.M., te Riele, H.J.J., Saouter, Y. (1998). New experimental results concerning the Goldbach conjecture. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054863

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    • DOI: https://doi.org/10.1007/BFb0054863

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    • Publisher Name: Springer, Berlin, Heidelberg

    • Print ISBN: 978-3-540-64657-0

    • Online ISBN: 978-3-540-69113-6

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