Integer parts of powers of quadratic units

Abstract:

Let $\alpha > 1$ be a unit in a quadratic field. The integer part of ${\alpha ^n}$, denoted $[{\alpha ^n}]$, is shown to be composite infinitely often. Provided $\alpha \ne (1 + \sqrt 5 )/2$, it is shown that the number of primes among $[\alpha ],[{\alpha ^2}], \ldots ,[{\alpha ^n}]$ is bounded by a function asymptotic to $c \cdot {\log ^2}n$, with $c = 1/(2\log 2 \cdot \log 3)$.