Some Systems of Diophantine Equations of the Tarry-Escott Type

A solution of (1.1) in which the a's merely form a permutation of the b's is called trivial. In what follows we are concerned with only non-trivial solutions.

We define ρ = ρ (k) as the least value ρ such that

a₁, a₂,..., a_ρ x b₁, b₂,.. ., b_ρ {x= 1, 2, . . ., k)

will have non-trivial solutions. The following easy theorem was first established by Bastein [1].