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I am somewhat surprised that the fact that it is indeed possible to generate the primes from a set of diophantine equations doesn't get more play:

Another approach might be to ask if there is a non-constant polynomial all of whose positive values (as the variables range in the set of non-negative integers) are all primes. Matijasevic showed this was possible in 1971 [Matijasevic71], and in 1976 Jones, Sato, Wada and Wiens gave the following explicit example of such a polynomial with 26 variables (and degree 25).

(k+2){1 – [wz+h+j–q]² – [(gk+2g+k+1)(h+j)+h–z]² – [2n+p+q+z–e]² – [16(k+1)³(k+2)(n+1)²+1–f²]² – [e³(e+2)(a+1)²+1–o²]² – [(a²–1)y²+1–x²]² – [16r²y⁴(a²–1)+1–u²]² – [((a+u²(u²–a))²–1)(n+4dy)² + 1 – (x+cu)²]² – [n+l+v–y]² – [(a²–1)l²+1–m²]² – [ai+k+1–l–i]² – [p+l(a–n–1)+b(2an+2a–n²–2n–2)–m]² – [q+y(a–p–1)+s(2ap+2a–p²–2p–2)–x]² – [z+pl(a–p)+t(2ap–p²–1)–pm]²}

(From the web page http://primes.utm.edu/glossary/page.php/MatijasevicPoly.html . You can find them broken out here at MathWorld.)