TITLE: Primes written as the sum of two squares
QUESTION [12 upvotes]: A well known result due to Fermat states that any prime $p \equiv 1 \pmod 4$ can be written as the sum of two squares,$\,$ $p = a^2 + b^2$ .$\,$ We can define a set $\,A\,$ by collecting together all the $a$'s and $b$'s  which arise in this way (ignoring multiplicity).$\,$ Examples include:$\;$ $5 = 1^2 + 2^2$,$\;$$13 = 2^2 + 3^2$,$\;$$17 = 1^2 + 4^2$,$\;$$29 = 2^2 + 5^2$,$\;$and $\,$$37 = 1^2 + 6^2$. $\;$From these examples we see that $\,A\,$ begins with $\;A = \,${$\,$$1,2,3,4,5,6, ...$$\,$}$\;$.
Questions: $\;$ 1) Does the set $\,A\,$ coincide with the set of all natural numbers?
$\qquad$$\qquad$$\quad$2) If not or if this is not known, is there at least a good lower bound for the density of$\qquad$$\qquad$$\quad$$\;$$\;$ the set$\,A\,$?
Thanks.

REPLY [3 votes]: 2) By looking at this derivation, there are asymptoticaly $2n^2/\log n$ Gaussian primes in the circle $|z|<n$, and so, in the square $0<\Re z,\Im z<n$ there are at least (asymptoticaly) $n^2/2\log n$ Gaussian primes. Hence, by pigeonhole principle, at least $n/2\log n$ abscissas out of first $n$ contain at least one Gaussian prime.
This is a very crude lower bound, but it's better than nothing.