Gaussian primes

Gaussian primes are Gaussian integers z = a+bi satisfying one of the following properties. 
1. If both a and b are nonzero then, a+bi is a Gaussian prime iff a^2+b^2 is an ordinary prime. 
2. If a = 0, then bi is a Gaussian prime iff |b| is an ordinary prime and |b|congruent3 (mod 4). 
3. If b = 0, then a is a Gaussian prime iff |a| is an ordinary prime and |a|congruent3 (mod 4). 
The above plot of the complex plane shows the Gaussian primes as filled squares.
The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ....

 The Gaussian primes with |a|, |b|<=5 are given by -5-4i, -5-2i, -5+2i, -5+4i, -4-5i, -4-i, -4+i, -4+5i, -3-2i, -3, -3+2i, -2-5i, -2-3i, -2-i, -2+i, -2+3i, -2+5i, -1-4i, -1-2i, -1-i, -1+i, -1+2i, -1+4i, -3 i, 3i, 1-4i, 1-2i, 1-i, 1+i, 1+2i, 1+4i, 2-5i, 2-3i, 2-i, 2+i, 2+3i, 2+5i, 3-2i, 3, 3+2i, 4-5i, 4-i, 4+i, 4+5i, 5-4i, 5-2i, 5+2i, 5+4i.