The main purpose of this paper is using the properties
of Gauss sums and the estimate for character sums to study the
properties of the primitive roots of
(an odd prime), and
prove that for any integers
and
, there
exists a primitive root
of
such that
is
also a primitive root of
, provide
large enough. Let
denotes the number of all primitive roots
of
such that
is also a primitive root of
. Then we
can also give an interesting
asymptotic formula for
.