Anton
Mosunov, Department
of
Pure
Mathematics,
University
of
Waterloo
“On
the
Representation
of
Integers
by
Binary
Forms
Defined
by
Means
of
the
Relation
(x
+
yi)n
=
Rn(x,
y)
+
Jn(x,
y)i”
Let
F
be
a
binary
form
with
integer
coefficients,
degree
d
≥
3
and
non-zero
discriminant.
Let
RF
(Z)
denote
the
number
of
integers
of
absolute
value
at
most
Z
which
are
represented
by
F.
In
2019
Stewart
and
Xiao
proved
that
RF
(Z)
∼
CFZ2/d
for
some
positive
number
CF
.
We
compute
CRn
and
CJn
for
the
binary
forms
Rn(x,
y)
and
Jn(x,
y)
defined
by
means
of
the
relation
(x
+
yi)n
=
Rn(x,
y)
+
Jn(x,
y)i,
where
the
variables
x
and
y
are
real.
MC
5479
and
Zoom
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