[Math] The “universal” diophantine equation

Question

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

1. All other diophantine equations (could be wrong on this)
2. 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.

Answer 1

Any Diophantine set can be enumerated, in the sense that there is a procedure that will list any given member of the set after a finite amount of time. In fact, Diophantine sets are precisely those which can be so listed: the recursively enumerable sets.

For the primes and many other Diophantine sets, more is true: the elements can be listed in order, since there are procedures for determining not only existence but also non-existence. These sets are called recursive or decidable.

You are right that finding a Diophantine representation for primes does not add much if anything to the study of primes. It serves, rather, to increase our understanding of Diophantine equations.