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Mathematics of Computation

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Empirical verification of the even Goldbach conjecture and computation of prime gaps up to $4\cdot 10^{18}$

Authors: Tomás Oliveira e Silva, Siegfried Herzog and Silvio Pardi
Journal: Math. Comp. 83 (2014), 2033-2060
MSC (2010): Primary 11A41, 11P32, 11N35; Secondary 11N05, 11Y55
Published electronically: November 18, 2013
MathSciNet review: 3194140
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Abstract: 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

Siegfried Herzog
Affiliation: Mont Alto Campus, The Pennsylvania State University, One Campus Drive, Mont Alto, Pennsylvania 17237

Silvio Pardi
Affiliation: INFN–Sezione di Napoli, Italy

Keywords: Goldbach conjecture, prime gaps, prime $k$-tuple conjecture
Received by editor(s): May 21, 2012
Received by editor(s) in revised form: December 6, 2012
Published electronically: November 18, 2013
Article copyright: © Copyright 2013 American Mathematical Society