There are certainly patterns. Try this:

Draw lines from the origin with slopes 1, 1/2, 1/3, 1/4, ...

A number, y=n, is prime if no line intersects y=n at an integer other than n.

You can also start out with a line of points, then repeatedly place a new line of points, translated up one unit and right one unit from the start of the last line and increase the distance between each point by one from the last line. [There are many other simple constructions and many repeating sub-units with definite patterns.]

           *         *
          *        *
         *       *   
        *      *      *
       *     *     *
      *    *    *    *
     *   *   *   *   *
    *  *  *  *  *  *  *
   * * * * * * * * * * 
  *********************

The primes are the columns with only two filled points.

http://divisorplot.com records more patterns.

I currently have a demo [0] up that lets you explore these drawings in projective spaces. [Note that the default scene is showing only a scaled portion of the underlying data.]

[0] http://blakelapierre.github.io/jsxbox/