@MoreAnonymous I agree with Misha's answer to your meta question that as stated your question is meaningless or ill-defined, because a finite-memory computer definitely cannot generate in any reasonable sense the sequence of prime numbers.
@MoreAnonymous: I also recommend that you study logic proper.
In particular, you say:
> I think intuitively I feel like my algorithm takes a sequence which has some "sense" behind it and creates a pattern/fractal out of it
Frankly speaking, this makes completely no sense.
By the way, I'm not speaking as a "more experienced user" in my above comments, but speaking from a perspective based on logic and computer science. If you have a contractive mapping, iterating it on any initial point will result in a sequence tending to some fixed point. This does not at all suggest that the initial point was special. Same with iterating your process on any initial sequence; whether the result has a 'pattern' or not does not imply the original had a 'pattern'.
@user21820 umm ... would it be fair to say that the algorithm generates patterns for a deterministic process rather than a random process?
I was thinking of a random process as: a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
I feel this is going in circles ... But u said, "you observing a pattern when you do X to Y does not imply that Y is special at all. It could be that X is special, or it could be that you are just seeking to find a pattern even if there is none. Don't underestimate that possibility."
We should be able to atleast find a counterexample if that is true
Now if you relax your question to be about Turing machines (not finite-memory machines), the answer to your question would be the mundane fact that many processes are not invertible and will 'collapse' many different inputs to the same output.
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a natural number -- n/2 if it is even, 3n+1 if it is odd.
The conjecture states that if this operation...
In this paper http://arxiv.org/abs/math/0602498 a sequence of integers is proposed, which, when started with $1$ begins like this:
$$1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, \dots$$
This is also the sequence A090822 at OEIS. The description there is somewhat better:
Gijswijt...
> Curiously, this paper explicitly states that the authors initially thought that no number greater than 4 appears in the sequence.
This is obviously an instance of what I said earlier, namely finding patterns when there may be none.
@MoreAnonymous I would still be incredibly unconvinced. The example I pointed you to had a counter-example at iteration 10^(10^28)...
@MoreAnonymous Different 'pattern' just means that your process does not have a single 'fixed-point', not that there is much information in the resulting sequence pertaining to the original.
@MoreAnonymous Of course; that's my claim, that a whole lot of processes like your will produce 'patterns' out from 'most' sequences, and randomness (whatever it means) is irrelevant.
In fact, such processes would work more on random sequences than on non-random ones. For instance try your process on the factorial sequence. Does it really produce any 'pattern'?
@MoreAnonymous: I have an example process, though I don't know how convincing it would be to you. Let the initial sequence be A[1][1..] and initialize A[0][0..] to zero, and A[k][0] = 0. For each iteration k and positive integer n, let A[k+1][n] = min(abs(A[k][n]−A[k][n−1]),abs(A[k][n]−A[k][n+1]),abs(A[k][n]−A[k−1][n])).
We can prove that for each positive integer n we have ( A[k][n] → 0 as k → ∞ ), but in many cases A[k] will never be the zero sequence no matter how large k is.
Also, if you try various initial sequences, there are various 'patterns' you can 'find' in the resulting array, and they have nothing to do with whether the initial sequence if random or not. Same like for your process.
The process I originally started with was simpler, namely just A[k+1][n] = min(abs(A[k][n]−A[k][n−1]),abs(A[k][n]−A[k][n+1])). You should try that one as well. I added the third term to force the process to have non-uniform convergence. I'll leave you to find the interesting behaviour in either of these processes.
In this paper http://arxiv.org/abs/math/0602498 a sequence of integers is proposed, which, when started with $1$ begins like this:
$$1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, \dots$$
This is also the sequence A090822 at OEIS. The description there is somewhat better:
Gijswijt...
