# examples of Gaussian primes

Even when we limit the real part^{} to the range 1 to 100 and the imaginary part to $i$ to $100i$, we come up with more than a thousand Gaussian primes^{}. Limiting the real part to 1 to 25 and the imaginary part to $i$ to $25i$ 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 $a+bi$ is prime then so is $a-bi$, $-a+bi$ and $-a-bi$. The list could be narrowed down even further by removing associates (e.g., $13+8i$ because $8+13i$ 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 $x+xi$ axis in the positive-positive quadrant and then reflecting them to the other side of that axis.

$1+i$, $1+2i$, $1+4i$, $1+6i$, $1+10i$, $1+14i$, $1+16i$, $1+20i$, $1+24i$

$2+i$, $2+3i$, $2+5i$, $2+7i$, $2+13i$, $2+15i$, $2+17i$

$3+2i$, $3+8i$, $3+10i$, $3+20i$

$4+i$, $4+5i$, $4+9i$, $4+11i$, $4+15i$, $4+21i$, $4+25i$

$5+2i$, $5+4i$, $5+6i$, $5+8i$, $5+16i$, $5+18i$, $5+22i$, $5+24i$

$6+i$, $6+5i$, $6+11i$, $6+19i$, $6+25i$

$7+2i$, $7+8i$, $7+10i$, $7+12i$, $7+18i$, $7+20i$

$8+3i$, $8+5i$, $8+7i$, $8+13i$, $8+17i$, $8+23i$

$9+4i$, $9+10i$, $9+14i$, $9+16i$

$10+i$, $10+3i$, $10+7i$, $10+9i$, $10+13i$, $10+17i$, $10+19i$, $10+21i$

$11+4i$, $11+6i$, $11+14i$, $11+20i$, $12+7i$, $12+13i$, $12+17i$, $12+23i$, $12+25i$

$13+2i$, $13+8i$, $13+10i$, $13+12i$, $13+20i$, $13+22i$

$14+i$, $14+9i$, $14+11i$, $14+15i$, $14+19i$, $14+25i$

$15+2i$, $15+4i$, $15+14i$, $15+22i$

$16+i$, $16+5i$, $16+9i$, $16+19i$, $16+25i$

$17+2i$, $17+8i$, $17+10i$, $17+12i$, $17+18i$, $17+22i$

$18+5i$, $18+7i$, $18+17i$, $18+23i$

$19+6i$, $19+10i$, $19+14i$, $19+16i$, $19+20i$, $19+24i$

$20+i$, $20+3i$, $20+7i$, $20+11i$, $20+13i$, $20+19i$, $20+23i$

$21+4i$, $21+10i$

$22+5i$, $22+13i$, $22+15i$, $22+17i$, $22+23i$, $22+25i$

$23+8i$, $23+12i$, $23+18i$, $23+20i$, $23+22i$

$24+i$, $24+5i$, $24+19i$, $24+25i$

$25+4i$, $25+6i$, $25+12i$, $25+14i$, $25+16i$, $25+22i$, $25+24i$

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 $2+0i$ shows that its real and imaginary parts are both even.

For a rational prime to be a Gaussian prime of the form $p+0i$, the real part has to be of the form $p=4n-1$. The ones in our sample range are 3, 7, 11, 19 and 23. As it happens, for $0+pi$ to be a Gaussian prime, $p$ also has to be of the form $4n-1$. The ones in our sample range are then $3i$, $7i$, $11i$, $19i$ and $23i$, 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.

Title | examples of Gaussian primes |
---|---|

Canonical name | ExamplesOfGaussianPrimes |

Date of creation | 2013-03-22 16:52:15 |

Last modified on | 2013-03-22 16:52:15 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 9 |

Author | PrimeFan (13766) |

Entry type | Example |

Classification | msc 11R04 |