J C Saunders, Pure Mathematics Department, University of Waterloo
“Sums of Digits in q-ary expansions”
Let
sq(n)
denote
the
sum
of
the
digits
of
a
number
n
in
base
q.
For
example,
s2(n)
represents
the
number
of
1s
in
the
binary
expansion
of
n.
In
1978,
Kenneth
B.
Stolarsky
showed
that
lim
inf
s2(n2)
=
0
n→∞
s2(n)
using
bounds
obtained
from
analytical
methods.
In
this
presentation,
we
verify
the
above
using
algebraic
methods
to
show
that
we
can
find
an
n
such
that
the
ratio
s2(n2)
s2
(n)
is
any
given
positive
rational
number.
We
will
also
see
that
a
few
other
intersting
results
about
the
analogous
ratio
involving
other
bases
and
other
powers,
including
fractional
exponents.