Additive basis
In additive number theory, an additive basis is a set formula_1 of natural numbers with the property that, for some finite number formula_2, every natural number can be expressed as a sum of formula_2 or fewer elements of formula_1. That is, the sumset of formula_2 copies of formula_1 consists of all natural numbers. The "order" or "degree" of an additive basis is the number formula_2. When the context of additive number theory is clear, an additive basis may simply be called a basis. An asymptotic additive basis is a set formula_1 for which all but finitely many natural  numbers can be expressed as a sum of formula_2 or fewer elements of formula_1. 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 for formula_2-sided polygons form an additive basis of order formula_2. Similarly, the solutions to Waring's problem imply that the formula_2th powers are an additive basis, although their order is more than formula_2. 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 states that, for any additive basis of order formula_2, the number of representations of the number formula_16 as a sum of formula_2 elements of the basis tends to infinity in the limit as formula_16 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 states that, for every formula_2, there exists an additive basis of order formula_2 whose  number of representations of each formula_16 is formula_22. 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 according to which the Schnirelmann density of a sum of two sequences is at least the sum of their Schnirelmann densities, unless their sum consists of all natural numbers. Thus, any sequence of Schnirelmann density formula_23 is an additive basis of order at most formula_24.