Is this version of Lagrange's four-square theorem true? Lagrange's four-square theorem states that any natural number $n$ can be represented as the sum of four integer squares.i.e. $n = a_1\times a_1 + a_2\times a_2 + a_3\times a_3 + a_4\times a_4$ > **Question:** Is there any natural number $k$ such that: $$\forall n\geq1~~\exists a_1,...,a_k\geq 0~;~~~~n=a_1^{a_1}+\cdots+a_k^{a_k}$$ > > In order to avoid triviality about small numbers we _define_ $0^0$ to be $0$ so in the case of $k>2$ we may represent $2$ as $1^1+1^1+0^0+...+0^0$. **Remark:** I think the answer should be negative because of the fast growth speed of exponentiation function which makes the building blocks ($a^a$ s) so _rare_ and this fact makes the necessary number $k$ larger and larger.

The answer is indeed negative.

As a simple proof, fix $k$ and consider the numbers from $1$ to $(2k)^{2k}$. Any $a_i$ in the representation of a number in this range must be at most $2k$. Therefore, there are at most $(2k+1)^{k}$ numbers $n$ in this range which can be represented as a sum $n=a_1^{a_1}+\cdots+a_k^{a_k}$. But as for $k>1$, $4k^2>2k+1$, it follows that $(2k)^{2k} > (2k+1)^k$, and there exists a number with no such representation.