In additive number theory, an additive basis of natural numbers. That is, the sumset consists of all natural numbers. The order or degree. When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis.

For example, by Lagrange's four-square theorem, the set of square numbers is an additive basis of order four, and more generally by the Fermat polygonal number theorem the polygonal numbers. Similarly, the solutions to Waring's problem. By Vinogradov's theorem, the prime numbers are an asymptotic additive basis of order at most four, and Goldbach's conjecture would imply that their order is three.

The unproven Erdős–Turán conjecture on additive bases goes to infinity. (More precisely, the number of representations has no finite supremum.) The related Erdős–Fuchs theorem states that the number of representations cannot be close to a linear function. The Erdős–Tetali theorem.

A theorem of Lev Schnirelmann states that any sequence with positive Schnirelmann density is an additive basis. This follows from a stronger theorem of Henry Mann.

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