Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition.
Principal objects of study include the sumset of two subsets A and B of elements from an abelian group G, and the h-fold sumset of A, The field is principally devoted to consideration of direct problems over (typically) the integers, that is, determining the structure of hA from the structure of A: for example, determining which elements can be represented as a sum from hA, where A is a fixed subset.
[1] Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2ℙ contains all even numbers greater than two, where ℙ is the set of primes) and Waring's problem (which asks how large must h be to guarantee that hAk contains all positive integers, where is the set of kth powers).
For example, Vinogradov proved that every sufficiently large odd number is the sum of three primes, and so every sufficiently large even integer is the sum of four primes.
Much current research in this area concerns properties of general asymptotic bases of finite order.