This is most probably asked here already in some other form (as it seems to be a very basic question) but since I have't found some question very similar to this one I feel obliged to ask (also because I do not know how to answer it at present and because this question appears in some of elementary researches I do).

Lagrange's four-square theorem guarantees that for every $m \in \mathbb N_0=\{0,1,2,...\}$ there is at least one quadruple of integers $(a_1(m),a_2(m),a_3(m),a_4(m))$ such that $m=\displaystyle\sum_{i=1}^{4}(a_i(m))^2$.

But, if the statement is slightly changed so to only allow quadruples that are subsets of $\mathbb N^4$ (instead of $\mathbb Z^4$ (or, equivalently, $\mathbb {N}_0^4$) then the statement is no longer true.

In other words, not every natural number is a sum of *exactly* four squares of naturals.

So, the set of all natural numbers (that is, elements of $\mathbb N=\{1,2,3,\dotsc\}$) that can be written as the sum of *exactly* four squares of natural numbers is not equal to all of $\mathbb N$.

If that exceptional set (the set of all natural numbers that cannot be represented as the sum of *exactly* four squares of naturals) is denoted as $E_4$, I have a basic question:

- How to characterize elements of $E_4$?

For example, is there a statement of the form "$w \in E_4$ if and only if $w$ is (or is not) of the form $a^k(bl+c)$", for some special choice (or choices) of $a$, $b$, $c$?

Or there is some other characterization?

**Edit**: If I did not misinterpret something, it seems that we have an answer here, so the question can be appropriately closed.