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

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\cdot 10^{18} . Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 ⋅ 10 26 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 k -tuple conjecture of Hardy and Littlewood (with an error that appears to be O ( t log ⁡ log ⁡ t ) O(\sqrt {t\log \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.