On Equal Sums of Like Powers and Related Problems

These pages are no longer maintained. They refer to results collected around 1994-1995 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 Jean-Charles 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
x₁^k + x₂^k + ... x_m^k = y₁^k + y₂^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 non-trivial 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 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 x₁^k + x₂^k + ... x_m^k = y₁^k + y₂^k + ... y_n^k = ... = z₁^k + z₂^k + ... z_q^k.
The Tarry-Escott problem: In this case, we are looking for solutions of the system of equations: x₁ + x₂ + ... x_m = y₁ + y₂ + ... y_m, x₁² + x₂² + ... x_m² = y₁² + y₂² + ... y_m², ... = ... x₁^k + x₂^k + ... x_m^k = y₁^k + y₂^k + ... y_m^k,
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 + ... x_m = y₁ + y₂ + ... y_m = ... = z₁ + z₂ + ... z_m, x₁² + x₂² + ... x_m² = y₁² + y₂² + ... y_m² = ... = z₁² + z₂² + ... z_m², ... = ... = ... = ... x₁^k + x₂^k + ... x_m^k = y₁^k + y₂^k + ... y_m^k = ... = z₁^k + z₂^k + ... z_m^k,