[FOM] Prime values of polynomials

Stephen G Simpson simpson at math.psu.edu
Thu Mar 6 17:26:08 EST 2008

My colleague Leonid 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.  Here Z is
the set of integers.  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/

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