Goldbach conjecture

A prime is a whole number greater than one whose only positive divisors are one and itself. Can every even whole number from four onward be written as the sum of two primes? For example, 4 = 2 + 2 and 12 = 5 + 7; using the same prime twice is allowed. The question asks for a rule covering every even target, however large.[1] [2]

A prime is a whole number greater than one whose only positive divisors are one and itself. Every even integer at least four is conjectured to be a sum of two primes. Does each target have at least one such representation? For $n=12$, the prime choices below twelve are $2,3,5,7,11$. Their complementary values $n-p$ are $10,9,7,5,1$; only the choices five and seven give prime complements. Define $R(n)$ to count ordered prime pairs $(p,q)$ with $p+q=n$. Then $R(12)=2$, counting $(5,7)$ and $(7,5)$, while $R(4)=1$ because $(2,2)$ is one ordered pair. The conjecture is $R(n)>0$ for every even $n\ge4$.

This count requires two tests to hold together: $p$ is prime and the reflected position $n-p$ is prime. It is therefore a correlation problem, not merely a question about the separate frequency of primes. The three-prime theorem, proved by Helfgott for odd integers greater than five, permits an extra summand and a different parity of target; it does not prove this two-prime assertion.[1] [2]

Many examples and partial theorems are known. None proves that every even number works.[1]

The binary assertion remains open. The August 2026 short-interval paper includes results conditional on the Riemann hypothesis; finding a Goldbach number in an interval does not cover every even integer there.[1]

Primes are the building blocks of multiplication: every whole number greater than one factors into primes. Goldbach asks whether adding just two of these building blocks reaches every even target as well. It connects the familiar way numbers are assembled by multiplication to a much less understood pattern in addition.[1] [2]

Each target reflects the prime set in a different way through the map p↦n−p. A general representation theorem must ensure that the prime set meets its reflected copy every time. This is a concrete link between prime distribution and additive number theory, which studies what numbers can be reached by sums. An interval theorem that finds one successful target measures a different kind of coverage.[1]