Parametrization of Pythagorean triples by a single triple of polynomials

It is well known that Pythagorean triples can be parametrized by two triples of polynomials with integer coefficients. We show that no single triple of polynomials with integer coefficients in any number of variables is sufficient, but that there exists a parametrization of Pythagorean triples by a single triple of integer-valued polynomials. The second author has recently studied polynomial parametrizations of solutions of Diophantine equations [8], and the first author has wondered if it were possible in some cases to parametrize by a k-tuple of integer-valued polynomials a solution set that is not parametrizable by a k-tuple of polynomials with integer coefficients [6]. Pythagorean triples provide an example that this is indeed so. We call a triple of integers (x, y, z) ∈ Z satisfying x + y = z a Pythagorean triple, and, if x, y, z > 0, a positive Pythagorean triple. It is well known that every Pythagorean triple is either of the form ( c(a − b), 2cab, c(a + b) )