I am interested into Diophantine equations, and I have few misunderstandings.
What exactly is a Diophantine representation of some set? It is some polynomial Diophantine equation, but the thing that I don't get, is how a single equation can be representative of the whole set (which by the way, can have an infinite number of elements).
More concretely, how does prime representation, that was given by Matijasevic, Robinson and Davis, generate every prime number?
Evidently they mean a polynomial in several variables and integer coefficients, such that taking all the variables as positive integers gives prime numbers when the value is positive, and all primes. In http://www.math.umd.edu/~laskow/Pubs/713/Diorepofprimes.pdf a polynomial with both modest number of variables and modest degree is given.
EDIT: Andre is right, about this article as well. It says: as the variables range over the non-negative integers, the set of positive values obtained is precisely the set of primes. Evidently, it can sometimes take on negative or zero values despite the variables being non-negative.
There is more to the story; the original reason Matijasevic got all the attention was his proof that there is no algorithm to decide whether a system of Diophantine equations has a solution. I was in college at the time; one of my professors was unhappy about the whole episode, the proof was much easier than anyone had expected, but he had not found it himself.
