Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

2013/11/18 by Tomás Oliveira e Silva, Siegfried Herzog, Silvio Pardi · 77 citations
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Algorithm #Analytic Number Theory Research #Annotation #Artificial intelligence #Combinatorics #Computer science #Conjecture #Goldbach's conjecture #Mathematics #Prime (order theory) #Prime number

paper · pdf · doi:10.1090/s0025-5718-2013-02787-1

published in Mathematics of Computation 83(288), 2033-2060 (American Mathematical Society (AMS))

crossref issued 2013/11/18 · crossref published 2013/11/18 · crossref published-online 2013/11/18 · openalex publication_date 2013/11/18 · crossref created 2013/11/18 · openalex created_date 2025/10/10 · crossref deposited 2026/04/21 · openalex updated_date 2026/08/06 · crossref indexed 2026/08/07

Abstract

This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 ⋅ 10 18 4⋅ 10¹⁸ . Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 ⋅ 10 26 8.37⋅ 10²⁶ . 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 k -tuple conjecture of Hardy and Littlewood (with an error that appears to be O ( t log ⁡ log ⁡ t ) O(√ tlog log t) , where t 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 1982 by Heath-Brown.

Citations

• Short effective intervals containing primes

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