[FOM] prime values of polynomials

[Stephen G Simpson]

Let Z be the set of integers.  My colleague L. Vaserstein here at the
Pennsylvania State University has studied sets of n-tuples of integers
of the form

  { (f_1(x_1,...,x_k), ..., f_n(x_1,dots,x_k)) | x_1,...,x_k in Z }

where f_1, ..., f_n are polynomials with coefficients in Z.
Vaserstein has shown that, for instance, the set of pairs of integers
which are relatively prime is of this form.  Also SL_2(Z), the set of
quadruples (a,b,c,d) in Z^4 such that ad - bc = 1, is of this form.
This answers a question of Skolem.  However, the set of prime numbers
is not of this form.

Vaserstein's paper is entitled "Polynomial parametrization for the
solutions of Diophantine equations and arithmetic groups" and is to
appear in Annals of Mathematics.

----

Name: Stephen G. Simpson

Affiliation: Professor of Mathematics, Pennsylvania State University

Research Interests: mathematical logic, foundations of mathematics

Web Page: http://www.math.psu.edu/simpson/