Michael Filaseta, University of South Carolina
“49598666989151226098104244512918”
If
p
is
a
prime
with
decimal
representation
dndn−1
.
.
.
d1d0,
then
a
theorem
of
A.
Cohn
implies
that
the
polynomial
f(x)
=
dnxn
+
dn−1xn−1
+
·
·
·
+
d1x
+
d0
is
irreducible.
One
can
view
this
result
as
following
from
the
fact
that
if
g(x)
∈
Z[x]
with
g(0)
=
1,
then
g(x)
has
a
root
in
the
disk
D={z∈C:|z|≤1}.
On
the
other
hand,
that
such
ag(x)
has
a
root
in
D
has
little
to
do
with
g(x)
having
integer
coefficients.
In
this
talk,
we
discuss
a
perhaps
surprising
result
about
the
location
of
a
zero
of
such
a
g(x)
that
makes
use
of
its
coefficients
being
in
Z
and
discuss
the
implications
this
has
on
generalizations
of
Cohn’s
theorem.
A
variety
of
open
problems
will
be
presented.
This
research
is
joint
work
with
a
now
former
student,
Sam
Gross.
Please note day.