What is... additive number theory ?

The central problem in additive number theory is to determine if a given set of integers is a basis of finite order.

Take for example the square numbers. According to Lagrange's theorem every integer can be represented as a sum of at most four squares. ( The number 7 for example, requires four squares: 7 = 2^2 + 1^2 + 1^2 + 1^2. ) In this example the set of squares is a basis of order 4.

Goldbach's conjecture is a famous problem in additive number theory.