First, some definitions:
Let A be a shortcut for an infinite sequence
(a (1), a (2), a (3), …)
just as B is an infinite series of positive integers
(b (1), b (2), b (3), …),
and C is an increasing infinite sequence of positive integers
(c (0), c (1), c (2), c (3), …). Call c (0) the initial value of C.
It is said that the pair A, B is a recursion pattern for C if for each
positive integer n we have
c (n) = a (n) * c (n-1) + b (n). Given a recursion pattern A, B and a
initial value for C, we can reconstruct all the sequence C.
Clearly, for any increasing infinite sequence C, we can find a couple A, B
which is a recursion pattern for C, simply by taking all the a (n) to
be equal to 1 and defining the b as required. If the sequence C is
increasing rapidly, there will be many A, B pairs that will be
recursion models for C.
In particular, for any increasing sequence of prime numbers, one can find
recursion models, many of them if the sequence grows rapidly.
Now the question: can a couple A, B be a recursion pattern for more than
a sequence of prime numbers? In other words, since A and B are a recursion
model, is it possible that there is more than an initial value for c (0)
which will lead to an infinite sequence of prime numbers?
If the answer is yes and we can prove it, I would wait for the proof
to be quite difficult. Maybe conditional proof would be possible.
On the other hand, it can be elementary to show that the answer is no.
This question was written by Moshe Newman.