See also here:
https://www.dima.to/blog/wp-content/uploads/2014/06/Primes.txtFor example, if you follow how the 0s move around they look like a rubber ball bouncing around. It's possible to follow its movement, even though there is some, uh, uncertainty about its exact position in some of the primes because it becomes "stretched", or sometimes it's even "contracted" at one side, but you can follow the path and see it bouncing consistently from left to right. With larger primes it changes a bit and the 1s seem to form some sort of subspaces in which the balls are contained, but they continue to behave the same otherwise.
The ways the 1s move is less chaotic and they essentially move to the left, bit by bit, sometimes jumping over neighbors, or a new 1 being shifted in from the right until they get destroyed after reaching the most significant digit on the left after which the number of digits increases. To some degree all this is simply about counting up, but following the pattern of movement of the bits it becomes somewhat predictable which single bit you have to move to get the next prime.
Then if you look sideways so that it becomes rotated at 90 degrees counterclockwise you can see something entirely different. It's a bit hard to see and I'm not really sure if or how it relates but If you ignore the least significant digit and instead follow the 1 on the top it starts looking kind of like a wave pattern...
There are exceptions here and there like skips of certain bits and some of the patterns become more er less noticeable for larger primes, but given any random prime it's easily possible to guess the next prime with a high chance of being correct. I have never seen anyone mention this and that seems surprising.
