A Comparison of Prime and Random Images
Random Images
Nice pictures, but what's the point? Well, the prime image just doesn't look random! Here we compare it to nine other images all produced using a similar algorithm - except randomly moving the coordinates instead of extracting the pixel direction from the prime numbers. In the final 'random' image, we have used the the first 10 million digits of pi (31415926)... notice how none of the random images are nearly as condensed as the prime one?
The Prime Number Theorem shows us that pi(x), that is, the number of primes there are less than or equal to x, can be quite accurately predicted with x/(log x - 1). Even if we use this to generate random values of about the same size as the primes, it doesn't make a difference to these images. The numbers are still random (and so is the coordinate movement).
So here are ten images (all can be enlarged to 1500x1500). The first eight are 100 million random pixels, plotted in the fashion I described on the 2d-page (some have strayed of the edge). The last one is the prime image. This is 5,761,455 (or ~ 100M/(log100M -1)) primes plotted as pixels in the same way.
