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# 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.

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## Corrections

some rational primes by pahio ✓

no associates! by pahio ✓

Parent hierarchy by CompositeFan ✓

## Comments

## Positivity only in R

One cannot speak of _positive_ Gaussian primes, meaning such as 25+24i, although the real and imaginary parts were positive. I don't know how we could call such numbers :)

## Re: Positivity only in R

One idea would be to name them after the quadrants of the complex plane.

## Re: Positivity only in R

What I meant to write the first time was "in the positive-positive quadrant." This is clunky and inelegant, but I think it gets the point across. Though it would be a valid question to ask if this includes or not the zero axes.

Anton had a few months ago suggested calling the quadrants Alpha, Beta, Gamma and Delta, in reference to the way the galaxy is divided in the Star Trek stories. But back then he wouldn't say if the Alpha Quadrant was the positive-positive, negative-positive, positive-negative or negative-negative.

## Re: Positivity only in R

I had in mind the the terminology "first quadrant", "second quadrant",

"third quadrant", "fourth quadrant" which one sees in analytic

geomerty texts. Of course, letters would work just as well as

numbers for labels of quadrants. As for which letter is which quadrant,

I would guess the obvious --- alpha first, beta second, gamma third,

delta fourth. In this connection, it might be worth noting that

leters serve as nummerals; where in a Latin document or inscription,

one mind find Roman numerals, in a Greek version, one would find

letters, perhaps with a mark to indicate that these are meant as

numerals. Eventually, it might be nice to have an entry on this

topic of Greek numerals and similar systems.

## picture

I remember having somewhere seen a plot of the Gaussian primes;

at any rate, it would not be too hard to generate and would

make a nice addition to this entry. All that would be needed

woudl be to take the list of numbers in this entry and fill in

the pixel at (m,n) whenever m + in is a Gaussian prime mnumber.

## Re: picture

With Mathematica I can plot Gaussian primes all day long if I wanted to. My problem would be getting the picture to show up here in PlanetMath. This is something that I have never been able to accomplish (e.g., MovingAverage).

## Re: Positivity only in R

The problem with analytic texts is that no one dares ask if everyone is on the same page, everyone pretends like they understand everything.

So, no one asks if indeed

4 + 7i is in first quadrant

-4 + 7i is in second quadrant

4 - 7i is in third quadrant

-4 - 7i is in fourth quadrant

or something else altogether. To take my own Star Trek analogy further, 4 + 7i is in the alpha quadrant because that's our own familiar quadrant. -4 - 7i is in the delta quadrant, where the Borg come from.

## Re: picture

To which graphics formats will Mathematica export output?

## Re: Mathematica export output

EPS, bitmap (BMP), Enhanced Meta File, Windows Meta File, Rich Text Format and Wave (WAV) (I haven't tried that last one, nor the meta files).

## Re: Mathematica export output

From what I understand, it should be possible to display EPS files

on PM. However, I don't know the details of how to do this nor

have done so myself, but people like Drini and Steve Cheng

would likely be able to offer expert advice on this point. Also

see the site document "Graphics and PlanetMath" in the document

section for more information.

## Re: Include graphics headaches

I looked at "Graphics and PlanetMath" in the document section and tried editing MovingAverage to match the example given for the PlanetMath logo. When I omitted the extension as shown, I got an error message saying the file couldn't be found. When I restored the file extension but omitted the begin and end figure so as to only leave the begin and end center, I got an error message saying that the file size of the graphic couldn't be determined. Oy vey!

## Re: Include graphics headaches

Actually I'm not even sure that PlanetMath even accepts PNG format for images. (Doing a little bit of googling, it seems to me that plain TeX/LaTeX doesn't accept PNG, and PM. pdfTeX and pdfLaTeX do seem to accept PNG, but as far as I know, PlanetMath does not use these.)

If you dig at the source of that graphics document, that PlanetMath logo image is indeed a EPS (Encapsulated PostScript) apparently converted from a bitmap image. Solution: do the same.

// Steve

## Re: Include graphics headaches

It worked! Thank you very much. With your advice and Anton's kludge, I finally got the illustrations for MovingAverage to show up. (Anton's kludge is uploading to GeoCities and then URL-grabbing from there. For some reason, if I upload from my computer, PM thinks the drive letter and semicolon is part of the filename, then it can show me the image but it can't show it to anyone else).