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# Gaussian primes

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Distribution the Gaussian primes

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

For example, neither 2 nor 5 are Gaussian primes, but since ${\displaystyle 2^{2}+5^{2}=29}$, which is a prime in ${\displaystyle \scriptstyle \mathbb {Z} \,}$,[1] then ${\displaystyle 2+5i}$ is a Gaussian prime, divisible only by itself (or its associates ${\displaystyle -2-5i}$, ${\displaystyle 5-2i}$, ${\displaystyle -5+2i}$) and 1 (or its associates ${\displaystyle -1}$, ${\displaystyle i}$ and ${\displaystyle -i}$) 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 ${\displaystyle \scriptstyle \mathbb {C} \,}$ given by ${\displaystyle \scriptstyle a^{2}+b^{2}\,=\,(a-bi)(a+bi)\,}$.

## Notes

1. Nor is 29 is a Gaussian prime either since ${\displaystyle (2+5i)(2-5i)=29}$.