Ideal solutions of the Tarry-Escott problem of degrees four and five and related Diophantine systems. (Q1419559)

Parametric ideal non-symmetric solutions in integers of the Tarry-Escott problem of degrees four and five are obtained; i.e., of the system of simultaneous equations \[ \sum_{i=1}^{k+1} a_i^r = \sum_{i=1}^{k+1} b_ i^r, \] \(r = 1, 2, \dotsc, k\) where \(k\) is 4 or 5. These non-symmetric solutions are used to obtain parametric solutions of the above Diophantine systems where \(r = 1, 2, \dotsc, k, k+2\), and \(k\) is 4 or 5.