Gaussian primes

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Distribution the Gaussian primes

A Gaussian prime is a non-unit Gaussian integer $\scriptstyle m+ni\,$ divisible only by its associates and by the units ($\scriptstyle 1,\,i,\,-1,\,-i\,$), and by no other Gaussian integers.

The Gaussian primes fall into one of three categories:

1. Gaussian integers with imaginary part zero and a prime real part $\scriptstyle m\,$ with $\scriptstyle |m|\,$ a real prime satisfying $\scriptstyle |m|\,\equiv \,3{\bmod {4}}\,$ (numbers of A002145 multiplied by $\scriptstyle 1\,$ or $\scriptstyle -1\,$).
2. Gaussian integers with real part zero and an imaginary part $\scriptstyle n\,$ with $\scriptstyle |n|\,$ a real prime satisfying $\scriptstyle |n|\,\equiv \,3{\bmod {4}}\,$ (numbers of A002145 multiplied by $\scriptstyle i\,$ or $\scriptstyle -i\,$).
3. Gaussian integers having both real and imaginary parts, and its complex norm is a real prime number in $\scriptstyle \mathbb {Z} \,$. (For example, given the numbers listed in A069003, the numbers $\scriptstyle n\pm a(n)i\,$ are Gaussian primes.)

For example, neither 2 nor 5 are Gaussian primes, but since $2^{2}+5^{2}=29$, which is a prime in $\scriptstyle \mathbb {Z} \,$, then $2+5i$ is a Gaussian prime, divisible only by itself (or its associates $-2-5i$, $5-2i$, $-5+2i$) and 1 (or its associates $-1$, $i$ and $-i$) but no other Gaussian integers.

It should be noted that although all Gaussian primes in category 1 above are in A000040, 2 and all primes congruent to 1 mod 4 (Pythagorean primes) are Gaussian composites, since they are the sum of two squares, with factorization in $\scriptstyle \mathbb {C} \,$ given by $\scriptstyle a^{2}+b^{2}\,=\,(a-bi)(a+bi)\,$.

See also

• Eisenstein primes