TITLE: Logical proposition for "Every positive integer can be written as the sum of 2 squares"
QUESTION [7 upvotes]: 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.

REPLY [7 votes]: 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.)

REPLY [5 votes]: 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.