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## The “universal” diophantine equation

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.

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Matiyasevich "Hilbert's 10th problem". See also en.wikipedia.org/wiki/Hilbert's_tenth_problem . The question is answered by Wikipedia. Voted to close. – Mark Sapir Nov 20 2011 at 15:13