## On Equal Sums of Like Powers and Related Problems

These pages provide information on a class of problems related to the diophantine equation
x1k + x2k + ... xmk = y1k + y2k + ... ynk, (1)
where xi, yj integers, and k is greater than or equal to zero. The problem is that of finding non-trivial solutions (that is, solutions where there is no xi equal to any yj 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 non-trivial 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 k-th powers of the integers of each set is the same. That is, we are looking for non-trivial solutions of the simultaneous equations
x1k + x2k + ... xmk = y1k + y2k + ... ynk = ... = z1k + z2k + ... zqk.

• The Tarry-Escott problem: In this case, we are looking for solutions of the system of equations:
x1 + x2 + ... xm = y1 + y2 + ... ym,
x12 + x22 + ... xm2 = y12 + y22 + ... ym2,
... = ...
x1k + x2k + ... xmk = y1k + y2k + ... ymk,

• The Prouhet-Tarry-Escott problem: In this case, we are looking for multiple solutions of the above system, i.e., solutions of the following system:
x1 + x2 + ... xm = y1 + y2 + ... ym = ... = z1 + z2 + ... zm,
x12 + x22 + ... xm2 = y12 + y22 + ... ym2 = ... = z12 + z22 + ... zm2,
... = ... = ... = ...
x1k + x2k + ... xmk = y1k + y2k + ... ymk = ... = z1k + z2k + ... zmk,