# Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive
integer s such that every natural number is the sum of at most $s$ $k$th powers of natural
numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth
powers, etc.) from here.

I ask: Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$, with $a_u\neq a_v \geq0 \; , \; \forall u\neq v$?

$g(k)$ being the minimum number $s$ of $k$th powers needed to represent all integers.
$14$ would be an example of a unique decomposition (into $0^2+1^2+2^2+3^2$).

Non-unique decompositions for $k=2$ can be constructed by letting $n=(a_0+x)^2+\sum_{j=1}^3 a_j^2$ and $a_0=x+\sum_{j=1}^3 a_j$. Robert gave a formula for cubes below.

So another question is: How can one test that a certain $n$ has a unique solution, or even better how can I calculate the number of respresentations? As Gerry points out in his comment, a prime $p=4k+1\;$ has a unique representation $p=a^2+b^2$ with $0<a<b\;$ (Thue’s Lemma).

Partial answer for $k=2$ (from here)

The sequence of positive integers whose representation as a sum of four squares is unique is:

1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224… (sequence A006431 in OEIS).

These integers consist of the seven odd numbers $1, 3, 5, 7, 11, 15, 23$ and all numbers of the form $2 × 4^k, 6 × 4^k$ or $14 × 4^k$.

Partial, therefore, because $23=1^2+2^2+2\times 3^2$ (answer, because $14\times 4^k$ is unique).

Another way to look at it, uses ${g(k)}$-dimensional vector spaces $V$ over $\mathbb N_0^+$ that are provided with an $k$-norm $||a||_k=\left( \sum_{j=1}^{g(k)} a_j^k \right)^{1/k}$ and the additional conditions that $a_u\neq a_v \geq0 \; , \; \forall u\neq v$.
Now there is a set of length-preserving operations $U_p$, that transforms vectors, from a defined range $\mathcal R (U_p)$. An example operation was shown above.
In this framework the question sounds like:

Which vectors of $V$ lie outside the union of all $\mathcal R (U_p)$?