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