Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.8 |

Department of Mathematics

Texas A&M University

Kingsville, TX 78363

USA

**Abstract:**

Let
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
*f*(*x*,*y*)=*axy*+*by*+*cx*+*d*,
,
*a*>1, *b*,*c*>0, assumes almost every positive integral value and almost every prime value when evaluated over the positive integers.

Periods exhibited in this expression are not assumed minimal. The analogous assertion holds for real quadratic units with prime trace and

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

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