(Non?)-uniqueness of sums of squares (I've had almost no exposure to number theory, so please keep answers as elementary as possible.)
Write $\mathbb{N} = \{0,1,2,3,\ldots\}$ for the natural numbers. Then every element of $\mathbb{N}$ can be expressed as a sum of squares. For example:
$$6 = 1+1+1+1+1+1$$
Usually, we can get away with fewer terms in our sum. For instance, in the preceeding example, we can get down from 6 terms to 3 by writing $6 = 4+1+1.$ So by the potency of $n \in \mathbb{N}$, let us mean the least $k \in \mathbb{N}$ such that $n$ can be expressed as a sum of $k$-many squares. Hence the potency of $6$ is $3.$

Question. Let $n \in \mathbb{N}$ denote a natural number with potency $k$. Suppose $x$ and $y$ are sequences of length $k$ in
  $\mathbb{N}$ such that $$n=\sum_{j = 1}^k x_j^2 = \sum_{j = 1}^k y_j^2.$$
Are the sequences $x$ and $y$ necessarily equal up to a reordering of their terms?

I'm guesssing not, but haven't been able to find a counterexample.
Further information.


*

*If $n$ is a prime of potency $2$, then the sequences $x$ and $y$ are equal up to reordering; see here.

*Here we learn that the corresponding statement for higher powers is false, since $1^3+5^3+5^3=2^3+3^3+6^3.$

There are many counterexamples, of which the smallest is $$5^2 + 5^2 = 7^2 + 1^2.$$
(I earlier stated that $25$ was the smallest counterexample, since $3^2+4^2 = 0^2+5^2$, but in your terminology, it has a potency of 1, not 2, so it is not a counterexample.)
Brahmagupta's identity shows that if $x$ and $y$ are each expressible as a sum of two squares, then $xy$ is (almost always) expressible as a sum of two squares in more than one way, because $$\begin{align}
(a^2 + b^2)(c^2 + d^2) & =
  (ac-bd)^2 + (ad+bc)^2 \\
&= (ac+bd)^2 + (ad-bc)^2
\end{align}$$
The counterexample of $50$ I gave above illustrates this, since $50 = 5\cdot 10 = (1^2+2^2)(1^2+3^2)$; you can use the Brahmagupta identity to find many similar counterexamples such as $(1^2+2^2)\cdot (2^2+3^2) = 1^2+8^2 = 4^2+7^2, $ etc.  By repeating this process, one can find numbers like $50\cdot 65 = 1^2 + 57^2 = 15^2+55^2 = 21^2 + 53^2 = 35^2 + 45^2 $ that have a potency of $2$ and are expressible as $a^2+b^2$ in arbitrarily many ways.
As André Nicolas points out, Lagrange proved that every positive integer has a potency of at most 4; the example of $7$ shows that this bound is attained.  The number 28 is the smallest number with a potency of 4 that can be expressed as a sum of four positive squares in multiple ways:
$$\begin{align}
28 & = 5^2 + 1^2 + 1^2 + 1^2 \\
   & = 3^2 + 3^2 + 3^2 + 1^2 \\
   & = 4^2 + 2^2 + 2^2 + 2^2
\end{align}$$
Other small examples include $31, 39, 47, 55, 60, $ and $63$, which is a sum of four positive squares in four ways.
