Is this a good expression of the claim using logical notation? Four Square Theorem: Every positive integer can be written as a sum of four integer squares. Expressed in logical notation : 
$$\forall n&gt;0 =a_{0}^{2} + a_{1}^{2} + a_{2}^{2} + a_{3}^{2}$$
In natural language: for all integers $n$, if $n &gt; 0$, then there are integers $a_0, a_1, a_2, a_3$ such that $n = a_0^2 + a_1^2 + a_2^2 + a_3^2$. Symbolically:
$$\forall n \in {\mathbb Z}: n &gt; 0 \to \exists a_0, a_1, a_2, a_3 \in {\mathbb Z}: n = a_0^2 + a_1^2 + a_2^2 + a_3^2.$$
If the context makes it clear that your talking about integers, you could also say
$$\forall n: n &gt; 0 \to \exists a_0, a_1, a_2, a_3: n = a_0^2 + a_1^2 + a_2^2 + a_3^2.$$
While I'm at this, since $0$ can be written as $0^2 + 0^2 + 0^2 + 0^2$ and since negative numbers are definitely not the sum of four squares, I'd quantify over the natural numbers and leave out the $n &gt; 0$ condition. That makes it a different, but equivalent statement.
$$\forall n \in {\mathbb N} \exists a_0, a_1, a_2, a_3 \in {\mathbb N}: n = a_0^2 + a_1^2 + a_2^2 + a_3^2.$$
I'd write it as 
$$\forall n &gt;0, \quad n = a_0^2+a_1^2+a_2^2+a_3^2$$