# Which basis orders [for the natural numbers] have been proven?

The set $A$ of nonnegative integers is called an additive basis of order $h$ if every nonnegative integer can be written as the sum of $h$ not necessarily distinct elements of $A$. For example, the famous Fermat-Cauchy-Lagrange "Four Square Theorem" is the statement that the squares are a basis of order four.

My question is: For which sets $A$ has the order $h(A)$ been proven?

Clearly, the polygonal numbers (i.e., triangular, square, pentagonal, etc.) are such sets, as are a [very] few of the smaller perfect powers such as cubes and fourth powers (q.v. Waring’s Problem). Kim’s paper on regular polytope numbers (http://www.ams.org/journals/proc/2003-131-01/S0002-9939-02-06710-2/S0002-9939-02-06710-2.pdf?q=on-regular-polytope-numbers) adds a few conjectured orders, but none are proven.

Is there a list somewhere of all sets of which the basis order has been proven?

• A good place to start is Melvyn Nathanson's book, Additive number theory. The classical bases. He has two other books that may also be of interest: One on what we now call additive combinatorics on the integers, including Freiman's theorem, and one on "elementary" methods in number theory. The latter includes a careful discussion of Waring's problems and some variants. By the way, Nathanson has worked on this topic for a long while, and he may be a reasonable person to contact asking for possible pointers or references. – Andrés E. Caicedo Jul 28 '14 at 15:02