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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Every odd number greater than $1$ is the sum of at most five primes
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by Terence Tao PDF
Math. Comp. 83 (2014), 997-1038 Request permission


We prove that every odd number $N$ greater than $1$ can be expressed as the sum of at most five primes, improving the result of Ramaré that every even natural number can be expressed as the sum of at most six primes. We follow the circle method of Hardy-Littlewood and Vinogradov, together with Vaughan’s identity; our additional techniques, which may be of interest for other Goldbach-type problems, include the use of smoothed exponential sums and optimisation of the Vaughan identity parameters to save or reduce some logarithmic losses, the use of multiple scales following some ideas of Bourgain, and the use of Montgomery’s uncertainty principle and the large sieve to improve the $L^2$ estimates on major arcs. Our argument relies on some previous numerical work, namely the verification of Richstein of the even Goldbach conjecture up to $4 \times 10^{14}$, and the verification of van de Lune and (independently) of Wedeniwski of the Riemann hypothesis up to height $3.29 \times 10^9$.
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Additional Information
  • Terence Tao
  • Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1596
  • MR Author ID: 361755
  • ORCID: 0000-0002-0140-7641
  • Email:
  • Received by editor(s): January 30, 2012
  • Received by editor(s) in revised form: July 3, 2012, and July 5, 2012
  • Published electronically: June 24, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 997-1038
  • MSC (2010): Primary 11P32
  • DOI:
  • MathSciNet review: 3143702