Cyclic Products of Purely Periodic Irrationals

Let (a0, · · · , ak−1) be a sequence of positive integers and m a positive integer. We prove that “almost every” real quadratic unit ǫ of norm (−1)k admits at leastm distinct factorizations into a product of purely periodic irrationals of the form ǫ = [a0; a1, . . . , ak−1, x, y]× [a1; a2, . . . , x, y, a0] × · · · × [y; a0, . . . , ak−1, x]. Periods exhibited in this expression are not assumed minimal. The analogous assertion holds for real quadratic units ǫ > 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.