Ulam's Spiral provides us with striking visual
evidence that there is some order in the distribution of prime numbers.
When graphing primes in this spiral fashion (explained below), mysterious
diagonal lines appear. Understanding these lines might be a lead
us to find good way of predicting the location of new prime numbers.
To this day, there exists no substancial explanation for this strange
phenomenon. The second image above has been run through an algorithm
that enhances significant diagonal lines.

This is how we plot pixels in a spiral fashion:

In the two first images at top, the prime numbers are plotted with
a white pixel. Those that also happen to be a Mersenne prime, are
plotted with a Red pixel. You can download the image and zoom into
it with some graphics package to see them in more detail.

Here the first 27 Mersenne primes are plotted by the spiral method...
Unfortunately, no obvious patterns can be seen..