[FOM] prime values of polynomials

Vaughan Pratt pratt at cs.stanford.edu
Wed Mar 19 03:32:44 EDT 2008

Many will be familiar with Pythagorean triples as an example of tuples 
so representable, namely as (m^2+n^2, m^2-n^2, 2mn).  One direction is 
an immediate consequence of Al-Karkhi's quarter-squares rule (p+q)^2 - 
(p-q)^2 = 4pq from around 1010 AD (but conjectured by David Smith to be 
"due to the Hindus") with p = m^2, q = n^2.  I found the rule handy 
recently in giving a cute proof of the equivalence of the Pythagorean 
formula for a right triangle and Heron's formula for the area of a 
triangle by factoring it as pq where p = s(s-a) and q =(s-b)(s-c) and 
applying Al-Karkhi (try it).


Stephen G Simpson wrote:
> 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/
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