Goldbach conjecture

Definitions and mathematical statement

Write $\mathcal P=\{2,3,5,7,\ldots\}$ for the prime positive integers. The binary Goldbach conjecture is the universal assertion \[ \forall N\in2\mathbb Z_{\ge2},\qquad \exists p,q\in\mathcal P\quad N=p+q. \] Here $2\mathbb Z_{\ge2}=\{4,6,8,\ldots\}$, and the primes $p,q$ need not be distinct. In particular $4=2+2$ and $6=3+3$ are admissible. For an even $N>4$, any representation uses two odd primes, since the only even prime is two.

To remove any convention about ordered pairs, define \[ r(N)=\#\{p\in\mathcal P:p\le N/2,\ N-p\in\mathcal P\}. \] The conjecture is equivalently $r(N)\ge1$ for every even $N\ge4$. It asks for existence, without a prescribed lower bound larger than one or an asymptotic formula for $r(N)$. A result for all sufficiently large even integers would still require the remaining finite range to be settled. Sums of three primes, or sums involving a semiprime, are different representations and do not satisfy the exact target. A counterexample must be an even integer at least four for which all possible prime summands up to $N/2$ fail.

Short English statement

Is every even integer at least four a sum of two primes, which may be equal?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3364 (OPG-706), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Goldbach conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/goldbach_conjecture.