There is a diophantine equation in some number (I think the minimum is now 9) of variables, that can be used to represent

- All other diophantine equations (could be wrong on this)
- Any particular set of numbers -- such as the primes

So to ask some questions around the consequence of this fact with another fact : the non-existence of a universal procedure for solving any diophantine.

Since no such procedure can exist, am I correct in concluding from these two facts that, at least as represented by a diophantine set, the primes can not be enumerated?

Or is the caveat that, maybe a particular procedure for solving a class or case of diophantine equations, of which the universal one mentioned above could be a member, exists and thus the primes could be enumerated by solving the universal one using such a yet-to-be-discovered specific method.

Also, one final question: Am I right in my feeling that construction of this universal diophantine is not really "adding any new insight" to the area of primes, but simply finding a way to represent some kind of computer or turing machine as a diophantine and program it.

If anyone would be so kind as to offer a simple explanation of the specific method of "programming" this diophantine or the constraints that actually give rise to this "universal" diophantine being able to encode the set of the primes, I would be grateful.