The only reason to use primes is to ensure that compound expressions are implicitly and uniquely related to their components. But with the magic of modern computers, this is unnecessary, as we have plenty of data storage and information representation implementations that simply store the relations explicitly (not to mention indexing them, enforcing uniqueness constraints, etc.).

In any case, the geometric (or perhaps you meant topological) properties of the objects are independent of the specific primes used, so "figuring out which primes to use initially" is a bit of a non sequitur.