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:
 * $\displaystyle \gamma \left({P}\right) = \prod_{i=1}^k p_i^{\beta \left({I_i}\right)}$

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.