# Seeking for an examples of non-trivial sets that can be used to generate all the natural numbers

First, let us remind ourselves of the Lagranges four square theorem which states that every natural number can be written (represented) as the sum of four integer squares. Since we have $(-a)^2=a^2$ we see that, in fact, the set $\{0\} \bigcup \{k^2: k \in \mathbb N\}$ can be used to generate all natural numbers by the process of adding four numbers from that set.

Let us now take the Triangular numbers as an example. On this linked page about triangular numbers there is a claim that "In 1796, German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including $T_0=0$)". So now we have that the set $\{0\} \bigcup \{\binom {n+1}{2}: n \in \mathbb N\}$ can be used to generate all natural numbers by the process of adding three (not necessarily different) numbers from that set.

One more example: If we suppose that Goldbachs conjecture is true then we can use the set $\mathbb P \bigcup \{1,0\}$ to generate all the natural numbers as the sum of at most three numbers from that set, because we can represent $1$ as $1+0$, $2$ as $1+1$, $3$ as $1+2$, any even natural number $2m\geq4$ as the sum of two primes and every odd natural number $2m+1\geq5$ as the sum of two primes and a number $1$.

So, in all these examples we had that some subset of $\mathbb Z$ was, in a certain sense, a generating set for the natural numbers in such a way that if we equipped that subset with some operation (adding four numbers from the set, or three) then we could generate all the natural numbers.

So, basically, I would like to know:

Are there any other non-trivial examples of sets which are subsets of $\mathbb Z$ and which can be equipped with some operation that can be applied no more than a given number of times so that that subset of $\mathbb Z$ together with that operation can be used to generate all the natural numbers?

It could be that my question is not clear enough but I do not know how to state it to be clearer.

Also, it is no problem to me if you give an answer which depends on usage of some unproven conjectures.

And, there is no need to give as an example Fermat polygonal number theorem because I am aware of it.

Thank you.

• We can use $k$-th powers. – André Nicolas May 1 '16 at 15:08
• @AndréNicolas How many of them for some $k$? – Farewell May 1 '16 at 15:29
• You can find information by googling Waring's Problem. Wikipedia will probably tell you all that you want. – André Nicolas May 1 '16 at 15:32