# Goldbach conjecture verification

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2021-09-17 15:00:16

The Goldbach conjecture is one of the oldest unsolved problems in number theory [1, problem C1]. In its modern form, it states that every even number larger than two can be expressed as a sum of two prime numbers.

Let n be an even number larger than two, and let n=p+q, with p and q prime numbers, p<=q, be a Goldbach partition of n. Let r(n) be the number of Goldbach partitions of n. The number of ways of writing n as a sum of two prime numbers, when the order of the two primes is important, is thus R(n)=2r(n) when n/2 is not a prime and is R(n)=2r(n)-1 when n/2 is a prime. The Goldbach conjecture states that r(n)>0, or, equivalently, that R(n)>0, for every even n larger than two.

In their famous memoir [2, conjecture A], Hardy and Littlewood conjectured that when n tends to infinity, R(n) tends asymptotically to (i.e., the ratio of the two functions tends to one)

The numerical evidence supporting this conjectured asymptotic formula is very strong. Up to 10^10, the Crandall-Pomerance formula does not deviate from R(n) by more than 40150, and up to 2^40 it does not deviate from R(n) by more than 401900.