Logical proposition for "Every positive integer can be written as the sum of 2 squares" I am to formulate the logical proposition for 

Every positive integer can be written as the sum of 2 squares (domain of integers)

One of the previous questions was

Formulate the logical proposition for the sentence "The number 100 can be written as the sum of 2 squares"  (domain of integers)

And my answer was $$ \exists x \exists y(x^2 + y^2 = 100)$$
So I thought to take the same approach with 

Every positive integer can be written as the sum of 2 squares

And came up with 
$$ \forall x \exists a \exists b (a^2 + b^2 = x)$$
But there's no where stating that $x$ is positive, so would
$$ \exists x \exists a \exists b (a^2 + b^2 = x)$$
be correct instead?
If someone could tell me where I went wrong or give me a hint I'd greatly appreciate the help, thank you very much.
 A: You can solve this question by using your previous answer,
$$P_{100}:= \exists\ x,y\ (x^2 + y^2 = 100)$$
where you replace $100$ by "any positive integer", let $z$:
$$\forall z>0\ P_z,$$ giving
$$\forall z>0\ \exists\ x,y\ (x^2 + y^2 = z).$$
($x,y,z\in\mathbb Z$ is left implicit.)
A: You can formulate this statement as:
$\large\forall_{x\in\mathbb{N}}\exists_{a,b\in\mathbb{N}}:x=a^2+b^2$

If you're worried about $0\in\mathbb{N}$ (which is a matter of definition), then you can use $\mathbb{Z^+}$ instead.
BTW, the statement itself is false.
A: Your second version is an a fortiori of what you're trying to express: there must exist such an $x$ since you "know" every $x$ has that form!
I think just $\forall x (x>0 \implies \exists a \exists b (a^2+b^2=x))$ will do. 
A: Why would you switch from "forall" to "exists" if you wanted to specify "$x$ is positive"?
You're going to want $$(\forall x > 0)(\exists a \exists b)(a^2+b^2 = x)$$
or, if your language doesn't let you formulate that, $$(\forall x)[x > 0 \to (\exists a \exists b)(a^2+b^2 = x)]$$
As an aside, the proposition is false: $3$ cannot be written as the sum of two squares. It is necessary and sufficient that primes $3 \pmod{4}$ appear only to even powers in the prime factorisation.
