examples of Gaussian primes

Even when we limit the real partMathworldPlanetmath to the range 1 to 100 and the imaginary part to i to 100i, we come up with more than a thousand Gaussian primesMathworldPlanetmath. 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 quadrantMathworldPlanetmath 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 ‘reflectionsMathworldPlanetmath’ of each other and give yet more axes of symmetryMathworldPlanetmath 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