Timothy Caley, Department of Pure Mathematics, University of Waterloo
"A new algorithm for the Prouhet-Tarry-Escott problem"
Abstract:
Given
natural
numbers
n
and
k,
the
Prouhet-Tarry-Escott
(PTE)
asks
for
integers
x_1,..,x_n
and
y_1,...,y_n
such
that
the
sums
of
the
first
k
powers
are
equal.
This
problem
has
connections
to
combinatorics
and
theoretical
computer
science,
as
well
as
to
other
areas
of
number
theory,
such
as
Waring's
problem.
The
most
interesting
case
is
when
k=n-1,
which
is
called
ideal.
A
major
open
problem
is
determining
whether
ideal
PTE
solutions
exist
for
a
given
n,
as
well
as
characterizing
those
that
do
exist.
Computational
techniques
have
been
used
to
search
for
PTE
solutions.
In
this
talk,
we
present
a
new
algorithm
to
find
PTE
solutions,
and
explain
how
the
results
yield
more
information
than
other
computational
searches
in
the
literature.