These pages are no longer maintained.
They refer to results collected around
19941995 using idle time on a network of SUN workstations. The most interesting results (ie, those that
were going
beyond what was then known) were
presented much later
as a poster at HPCNE'98 (Published by Springer in Lecture Notes in Computer
Science, vol. 1401, Apr. 1998),
which is available here.
A report (ref.5 in the paper) containing all the results found during this exercise
is given here (sorry it is a bit
messy!).
For the current status, the definitive place to check is
JeanCharles Meyrignac's site.
On Equal Sums of Like Powers and Related Problems
These pages provide information on a class of problems
related to the diophantine equation
x_{1}^{k} +
x_{2}^{k} + ...
x_{m}^{k} =
y_{1}^{k} +
y_{2}^{k} + ...
y_{n}^{k}, (1)
where x_{i}, y_{j} integers, and k is greater than
or equal to zero. The problem is that of finding nontrivial solutions
(that is, solutions where there is no
x_{i} equal to any y_{j} and vice
versa) of the above equation.
Equations of this form have a long history. It can be seen that
for m=1, n=2, Equation (1) has no solutions
for k > 2 according to Fermat's Last Theorem. In 1769, this
was generalised by Euler who conjectured that, for m=1, there
are no solutions with k less than n. The first counterexample was found
in 1966.

Equal
Sums of Like Powers:
In this case we are looking for nontrivial solutions of the above
equation for given k, m, n.

Multiple Equal Sums of Like Powers:
In this case we are looking for more than two sets of integers
that satisfy (1) and where the sum of the kth
powers of the integers of each set is the same. That is, we
are looking for nontrivial solutions of the simultaneous equations
x_{1}^{k} +
x_{2}^{k} + ...
x_{m}^{k} =
y_{1}^{k} +
y_{2}^{k} + ...
y_{n}^{k} = ... =
z_{1}^{k} +
z_{2}^{k} + ...
z_{q}^{k}.

The TarryEscott problem:
In this case, we are looking for solutions of the system of
equations:
x_{1} +
x_{2} + ...
x_{m} =
y_{1} +
y_{2} + ...
y_{m},
x_{1}^{2} +
x_{2}^{2} + ...
x_{m}^{2} =
y_{1}^{2} +
y_{2}^{2} + ...
y_{m}^{2},
... = ...
x_{1}^{k} +
x_{2}^{k} + ...
x_{m}^{k} =
y_{1}^{k} +
y_{2}^{k} + ...
y_{m}^{k},

The ProuhetTarryEscott problem:
In this case, we are looking for multiple solutions of the above
system, i.e., solutions of the following system:
x_{1} +
x_{2} + ...
x_{m} =
y_{1} +
y_{2} + ...
y_{m} = ... =
z_{1} +
z_{2} + ...
z_{m},
x_{1}^{2} +
x_{2}^{2} + ...
x_{m}^{2} =
y_{1}^{2} +
y_{2}^{2} + ...
y_{m}^{2} = ... =
z_{1}^{2} +
z_{2}^{2} + ...
z_{m}^{2},
... = ... = ... = ...
x_{1}^{k} +
x_{2}^{k} + ...
x_{m}^{k} =
y_{1}^{k} +
y_{2}^{k} + ...
y_{m}^{k} = ... =
z_{1}^{k} +
z_{2}^{k} + ...
z_{m}^{k},
© Rizos Sakellariou, 1998.
rizos@cs.man.ac.uk