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Gaussian Primes With Real Part To And Imaginary Part I To
Even
when we limit
the real part
to the range
1 to 100 and the imaginary part
to 
to 
, we come up with more than a thousand
Gaussian primes. Limiting the real part to 1 to 25 and the imaginary part to 
to 
gives us a list approximately a quarter of the size.
It makes sense to limit the listing to the positive-positive quadrant
of the complex plane, since if 
is prime
then so is 
, 
and 
. The list could be narrowed down even further by removing associates
(e.g., 
because 
appears first), but they have been left in. Thus, assuming the list has no mistakes, plotting these values should give the same result as plotting all Gaussian primes under (or over) the 
axis
in the positive-positive quadrant and then reflecting them to the other side
of that axis.
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As you may notice from the listing above, the real
and the imaginary parts must be of different parity. Thus, 2, which is a prime among the real primes, is not a prime among the Gaussian primes, since its complex
notation 
shows that its real and imaginary parts are both even.
For a rational prime
to be a Gaussian prime of the form 
, the real part has to be of the form
. The ones in our sample range are 3, 7, 11, 19 and 23. As it happens, for 
to be a Gaussian prime, 
also has to be of the form 
. The ones in our sample range are then 
, 
, 
, 
and 
, which ought to look a lot like the previous listing because they are the associates of the Gaussian primes with no imaginary part. Thus, the 0 axes are `reflections' of each other and give yet more axes of symmetry
of the pattern.