Goldbach's conjecture

A prime is an integer $p>1$ whose only positive divisors are $1$ and $p$.

Conjecture (strong, or binary, Goldbach conjecture). Every even integer $n>2$ is the sum of two primes:

$$\forall n\in\mathbb N,\quad \bigl(n>2 \ \text{and}\ 2\mid n\bigr)\ \Longrightarrow\ \exists\,p,q \text{ prime with } n=p+q .$$

The two primes need not be distinct, so $4=2+2$ and $6=3+3$ count. The problem is to decide the statement. A solution may either prove it for every even $n>2$, or disprove it by exhibiting an even $n>2$ that is not a sum of two primes. Both directions are open. The statement has been checked by computer for all even $n\le4\cdot10^{18}$ [5], so any counterexample is larger than that. [1, 8]

A standard quantitative form counts representations with logarithmic weights,

$$r(N)=\sum_{\substack{p+q=N\\ p,q\ \text{prime}}}\log p\,\log q .$$

Hardy and Littlewood conjectured that for even $N\to\infty$,

$$r(N)\sim 2C_2\,N\prod_{\substack{p\mid N\\ p>2}}\frac{p-1}{p-2},\qquad C_2=\prod_{p>2}\Bigl(1-\frac{1}{(p-1)^2}\Bigr),$$

where $C_2\approx0.6602$ is the twin prime constant [2]. This asymptotic is stronger than the conjecture and is not required for it.

Background

Goldbach raised the question in a letter to Euler dated 7 June 1742 [1]. Euler's reply recast it as the claim that every even number greater than $2$ is a sum of two primes. The weaker ternary form asks that every odd $n>5$ be a sum of three primes. The binary statement implies the ternary one, since $n-3$ is even.

Hardy and Littlewood developed the circle method for additive problems with primes and used it to derive the conjectural asymptotic above [2]. The circle method handles three primes well but fails for two. With two primes the minor arcs cannot be bounded by the available estimates. Sieve methods meet the parity problem, which prevents them from distinguishing integers with an odd number of prime factors from those with an even number. For these reasons the binary problem is regarded as much harder than the ternary one. It is one of Landau's four problems on primes, posed at the 1912 International Congress of Mathematicians. [8]