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.
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