OP-0007OpenNumber theory

The binary Goldbach conjecture

Is every even integer greater than 2 a sum of two primes?

MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Prove or disprove: every even integer $n > 2$ can be written as $n = p + q$ with $p, q$ prime.

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Background

Stated in a 1742 letter from Christian Goldbach to Euler. The ternary version — every odd number greater than 5 is a sum of three primes — was proved unconditionally by Helfgott in 2013, building on the circle method and extensive computation.

Known partial results

Verified for all even $n$ up to $4 \times 10^{18}$ .
Chen (1973): every sufficiently large even number is the sum of a prime and a number with at most two prime factors.
Montgomery–Vaughan: the number of even $n \le x$ not representable is $O(x^{1-\delta})$ for some $\delta > 0$ .
The ternary case is settled, which does not imply the binary case; the circle method's minor arcs are far harder with two variables.

Recorded progress

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