Cyclic Products of Purely Periodic Irrationals
C. R. Carroll
Department of Mathematics
Texas A&M University
Kingsville, TX 78363
be a sequence of positive integers and m
a positive integer. We prove that "almost every" real quadratic unit
of norm (-1)k
admits at least m
distinct factorizations into a product of purely periodic irrationals of the form
Periods exhibited in this expression are not assumed minimal. The analogous assertion holds for real quadratic units
with prime trace and m
=1. The proofs are based on the fact that an integral polynomial map of the form
>0, assumes almost every positive integral value and almost every prime value when evaluated over the positive integers.
Full version: pdf,
Received May 3 2013;
revised version received March 21 2014.
Published in Journal of Integer Sequences, March 22 2014.
Journal of Integer Sequences home page