Unique Code for URM Program

Theorem
Any URM program can be assigned a unique code number.

Proof
Let $$\mathbf{P}$$ be the set of all URM programs.

Let $$P \in \mathbf{P}$$ be a URM program with $$k$$ basic instructions:

We define the mapping $$\gamma: \mathbf{P} \to \N$$ as follows:
 * $$\gamma \left({P}\right) = \prod_{i=1}^k p_i^{\beta \left({I_i}\right)}$$

where:
 * $$p_i$$ is the $$i$$th prime number;
 * $$\beta \left({I_i}\right)$$ is the unique code for instruction $i$.

Hence it follows from the Fundamental Theorem of Arithmetic that $$\gamma$$ is uniquely specified for any given URM program.

Thus $$\gamma$$ is an injection.

For a given $$P$$, the number $$\gamma \left({P}\right)$$ is referred to as the code number of $$P$$.

Does Not Code
Not every $$e \in \N$$ is the code number of a URM program.

If $$e$$ is not the code of any URM program, we say that $$e$$ does not code a URM program.

Note
The coding scheme for $$\mathbf{P}$$ is not unique.

This particular scheme lends itself especially to number-theoretical analysis techniques.