Jeffrey Shallit, David R. Cheriton School of Computer Science, University of Waterloo
"Waring's theorem for binary k'th powers and palindromes"
Recently
there
has
been
some
interest
in
analogues
of
Waring's
theorem
for
other
kinds
of
sets
analogous
to
powers
of
integers. For
example,
Banks
proved
that
every
natural
number
is
the
sum
of
at
most
49 numbers
whose
base10 expansion
is
a
palindrome, and
this
was
improved
to
3 for
all
bases
b
≥
5 by
Cilleruelo,
Luca,
and
Baster.
In
this
talk
I
will
discuss
some
recent
results
about
palindromes in
bases
2,
3,
4 and
binary
k'th
powers.
(A
binary
k'th
power is
an
integer
whose
base2 expansion
consists
of
k consecutive
identical
blocks.)
In
particular,
I'll
show
that
for
all
k
≥
1 there
exists
a
constant
W(k) such
that
every
sufficiently
large multiple
of
gcd(k,2^{k}

1)$
is
the
sum
of
W(k) binary
k'th powers.
This
is
joint
work
with
Aayush
Rajasekaran,
Dirk
Nowotka, Parthasarathy
Madhusudan,
Tim
Smith,
Daniel
Kane,
and
Carlo
Sanna.
MC 5417