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Cyclic Products of Purely Periodic Irrationals C. R. Carroll
Department of Mathematics
Texas A&M University
Kingsville, TX 78363
USA
maito:c-carroll@tamuk.educ-carroll@tamuk.edu

in

Abstract:

Let $\left(a_{0}, \dotsb ,a_{k-1} \right)$ be a sequence of positive integers and m a positive integer. We prove that ``almost every'' real quadratic unit $\epsilon$ of norm (-1)^k admits at least m distinct factorizations into a product of purely periodic irrationals of the form

$${\epsilon= [\overline{a_{0}; a_{1}, \ldots ,a_{k-1},x,y}]\, \... ...es \cdots \times \,{[\overline{y;a_{0}, \ldots, a_{k-1},x}] }.$$

Periods exhibited in this expression are not assumed minimal. The analogous assertion holds for real quadratic units $\epsilon>1$ with prime trace and m=1. The proofs are based on the fact that an integral polynomial map of the form f(x,y)=axy+by+cx+d, $\gcd(a,bc)=1$, a>1, b,c>0, assumes almost every positive integral value and almost every prime value when evaluated over the positive integers.