Goldbachtype

Goldbachtype refers to problems and conjectures in additive number theory that ask whether integers can be expressed as sums of members of a specified set, most commonly the set of prime numbers. The phrase categorizes statements of Goldbach-like form, often framed as: does every sufficiently large integer n belong to the 2-fold sumset S+S, where S is the chosen set (for Goldbach-type problems, S is the primes)?

In the classical Goldbach problem, the set S is the primes, and the question is whether every even integer greater than 2 can be written as p+q with primes p and q. Such questions are sometimes described as Goldbach-type or Goldbach-type conjectures or results. The broader framework asks: given a set S with certain density or arithmetic structure, does every sufficiently large integer n belong to S+S?

Examples and status: The strong Goldbach conjecture asserts this for the primes for all even n > 2 and remains unproven. The weak Goldbach conjecture asserts every odd n > 5 is a sum of three primes; it was proven by Harald Helfgott in 2013–2014. Related results include Vinogradov’s theorem proving that every sufficiently large odd number is a sum of three primes, and various partial results where the number of primes or the size of the summands is constrained, or where the problem is restricted to primes in arithmetic progressions or to "almost primes" (numbers with few prime factors).

Methods and variants: Key tools include analytic techniques and the circle method, sieve methods, and estimates for primes in arithmetic progressions. Variants study representations with summands from dense subsets, or two numbers in prescribed residue classes, or the notion of a set being a basis of order two.

See also: Goldbach's conjecture; additive number theory; circle method; Vinogradov's theorem; Helfgott.