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OpenNumber Theory·1742

Goldbach's Conjecture

Every even integer greater than 2 should be expressible as a sum of two primes.

Formula

Strong conjecture
∀ n≥2,2n=p+q(p,q prime)\forall\, n\ge 2,\quad 2n = p + q\qquad (p,q\text{ prime})

Every even integer greater than 2 can be written as the sum of two primes.

Weak (ternary) conjecture (proved 2013)
∀ odd m>5,m=p1+p2+p3(pi prime)\forall\text{ odd } m > 5,\quad m = p_1 + p_2 + p_3\qquad (p_i\text{ prime})

Every odd integer greater than 5 is the sum of three primes. Proved by Helfgott in 2013.

Chen's theorem (1966)
2n=p+q,q prime or q=p1p2for all large n2n = p + q,\quad q\text{ prime or }q = p_1 p_2\quad\text{for all large }n

Every sufficiently large even integer is the sum of a prime and a number with at most two prime factors.

Summary

Goldbach's conjecture originated in a June 7, 1742 letter from Christian Goldbach to Leonhard Euler. In its modern form, it asserts that every even integer greater than 2 is the sum of two primes. The weak (ternary) version -- every odd integer greater than 5 is the sum of three primes -- was proved by Harald Helfgott in 2013. Chen Jingrun's 1966 theorem showed every sufficiently large even integer is the sum of a prime and a product of at most two primes. The strong conjecture has been computationally verified for all even integers up to 4 * 10^18 by Oliveira e Silva.

Sources

The ternary Goldbach conjecture is true

arXiv:1312.77482013

Major arcs for Goldbach's problem

arXiv:1305.28972013

On the representation of a large even integer as the sum of a prime and the product of at most two primes

1973

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