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Every odd number greater than $ 1$ is the sum of at most five primes


Author: Terence Tao
Journal: Math. Comp. 83 (2014), 997-1038
MSC (2010): Primary 11P32
DOI: https://doi.org/10.1090/S0025-5718-2013-02733-0
Published electronically: June 24, 2013
MathSciNet review: 3143702
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Abstract: 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
Email: tao@math.ucla.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02733-0
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.