Functions ℕ→ℕ from F1

Question

Which functions $\mathbb N\to\mathbb N$ can be represented by a term in the free rieg $F_1$ on one generator?

Here a term is built from $0,1,x$ using addition, multiplication, and binary exponentiation. Evaluating $x=n$ in the natural-number rieg gives its function, with $0^0=1$.

Two distinct tasks

1. Characterize the functions obtained by this evaluation.
2. Determine when two terms give the same function on $\mathbb N$.

The second question must distinguish equality of functions on this particular rieg from equality forced by the axioms in the free rieg.

Next step

Identify a property shared by all representable functions and prove it by induction on terms, or exhibit a function together with a rigorous nonrepresentability argument. A table of small inputs alone cannot establish representability or equality for all natural numbers.

Original problem 7.0.2.

@misc{sumire:2609.00007:4:v3,
  author = {hora},
  title = {{Functions ℕ→ℕ from F1}},
  year = {2026},
  note = {SUMIRE:2609.00007/4, v3},
  url = {https://sumire.aiformath.jp/p/2609.00007/n/4/v/3}
}