Weak Goldbach Conjecture

The weak Goldbach conjecture, also called the ternary Goldbach conjecture or three-primes problem, is the statement that every odd number is a sum of three primes, with repetitions allowed. It is now a theorem, proved by Helfgott (2013, 2014). For example, , , and . In the stronger formulation established by the proof, every odd number is a sum of three odd primes. The distinction at 7 is necessary because the smallest odd prime is 3.

The adjective "weak" records its relation to the strong, or binary, Goldbach conjecture, which asserts that every even number is a sum of two primes. If that assertion holds, then for each odd number , apply it to the even number and add 3 to obtain a three-prime representation of . The proof of the weak conjecture does not settle the strong Goldbach conjecture. Both problems arose from the correspondence of Goldbach and Euler in 1742 (Helfgott 2014).

Vinogradov's theorem established the result for every sufficiently large odd number (Vinogradov 1937). Helfgott's proof uses the circle method to bound the number of three-prime representations and show that it is positive for every odd number (Helfgott 2014). The remaining finite range is covered by computation. Helfgott and Platt (2013) verified the three-prime statement through . The analytic and computational ranges overlap, yielding a proof for all odd numbers greater than 5. This is an unconditional result and does not assume the generalized Riemann hypothesis.

A consequence is that every even number is a sum of four primes, since is an odd number to which the theorem applies. Together with the smaller cases, every integer greater than 1 is therefore a sum of at most four primes.