Cyclic Products of Purely Periodic Irrationals

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

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

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.

Received May 3 2013; revised version received March 21 2014. Published in Journal of Integer Sequences, March 22 2014.

Cyclic Products of Purely Periodic Irrationals

C. R. Carroll Department of Mathematics Texas A&M University Kingsville, TX 78363 USA

C. R. Carroll
Department of Mathematics
Texas A&M University
Kingsville, TX 78363
USA