The Cartesian Way
Half a parabola

After understanding that a prime is an integer that does not fall on the integer solutions of y(x) = x2 + b * x, for all values of b and x > 1 I thought it would be interesting to look at this in the cartesian plane.... the X and Y axis rather than integers arranged on an integer number spiral.

Whilst looking at a parbola for integer solutions the resulting graph of the parabola in the Cartesian plane is not continuous... probably easier to explain with images.

Below is an image of the postive solultions for y(x) = x2 while considered as a continuous function.


y(x) = x2 when considered as a continuous function

Below is an image of the postive solultions for y(x) = x2 while considered as a discrete function.


y(x) = x2 when considered as a discrete function

The difference is that in the continuous function we are considering all possible value of x. In the discrete function we are considering only the integer values of x.

In the graph of the continuous function you can pick a point anywhere along the red line and determine the values for x and y... even if x is 1.1567 (for example). Well... that's not quite the case with my graph because the resolution is not so good... but you get the idea.

The equation y(x) = x2 creates a parbola in the cartesian plane. The solution to the equation y(x) = x2  gives both a positive and a negative result... because when you square a negative number you get a positive number, so going in the reverse and taking the square root means that you must cater for both possibilities... that the number was negative and squared, or the number was positive and squared. -2 * -2 gives the same result as 2 * 2.

When interested in looking at the positive integers we are only concerned with the postive value of the solution for y(x) = x2... consequently we end up with half a parabola.

Below is zoom (it's a bit hard to see those red dots in the above image!) into an image that shows the positive solutions for a  y(x) = x2 when considering the intger values of x only (discrete function).

 

So I went further with this and layered the iterations of y(x) = x2 + b * x, incrementing the value of b (which is b = 0 for the above examples... giving y(x) = x2).

Below is an image of the resulting layering, showing the image up to y = 2000. That means a looooong image...


 

Let's zoom into it a little bit... zooming into the bottom:

This image can tell us where the primes are (not suprisingly since we're doing the same thing that we did to generate the Ulam spiral with  y(x) = x2 + b * x)

The primes are those values of Y that do not have any pixel marked. Below is an image clarifying this:

 

Below is a animation of the long image above, rotated and animated.... 

Copyright © 2007 - H Rudd