Here is a quick experiment to show that the Ulam spiral indeed appears to have more prominent diagonal lines than a spiral of randomly chosen odd numbers even when the dimensions and the number of points in both spirals are same: https://github.com/mycask/ulam-vs-random

I have plotted a 99x99 grid in this experiment. One can plot larger grids to see that the difference between the two grids (the Ulam spiral and the random spiral) becomes more apparent with larger sizes.

Update: After I made these text-plots, I found that Wikipedia has much better graphical plots that demonstrates the difference:

- Ulam spiral: https://en.wikipedia.org/wiki/File:Ulam_1.png

- Random spiral: https://en.wikipedia.org/wiki/File:Randomly_black_odd_number...