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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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Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$
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by Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi PDF
Math. Comp. 83 (2014), 2033-2060 Request permission


This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than $4\cdot 10^{18}$. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to $8.37\cdot 10^{26}$. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime $k$-tuple conjecture of Hardy and Littlewood (with an error that appears to be $O(\sqrt {t\log \log t})$, where $t$ is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in $1982$ by Heath-Brown.
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Additional Information
  • Tomás Oliveira e Silva
  • Affiliation: Departamento de Electrónica, Telecomunicações e Informática / IEETA, Universidade de Aveiro, Portugal
  • ORCID: 0000-0002-8878-3219
  • Email:
  • Siegfried Herzog
  • Affiliation: Mont Alto Campus, The Pennsylvania State University, One Campus Drive, Mont Alto, Pennsylvania 17237
  • Email:
  • Silvio Pardi
  • Affiliation: INFN–Sezione di Napoli, Italy
  • Email:
  • Received by editor(s): May 21, 2012
  • Received by editor(s) in revised form: December 6, 2012
  • Published electronically: November 18, 2013
  • © Copyright 2013 American Mathematical Society
  • Journal: Math. Comp. 83 (2014), 2033-2060
  • MSC (2010): Primary 11A41, 11P32, 11N35; Secondary 11N05, 11Y55
  • DOI:
  • MathSciNet review: 3194140