Ideal solutions of the Tarry-Escott Problem of degree seven

Abstract

In this paper we obtain four new parametric ideal solutions of the Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine equations, ${\sum }_{i=1}^8{x}_i^r={\sum }_{i=1}^8{y}_i^r,r=1,2,\dots ,7$. While all the known parametric solutions of the problem, with one exception, are given by polynomials of degrees $\ge 5$, the solutions obtained in this paper are given by quartic polynomials, and are thus simpler than almost all of the known solutions.

6 pages