On the number of solutions of the Tarry-Escott problem of degree two and the related problem over some finite fields (Q484576)

Let \(n\) and \(k\) be positive integers. A solution of the Tarry-Escott problem of size \(n\) and degree \(k\) consists of two distinct sets of integers \(\left\{a_1,a_2, \ldots, a_n\right\}\) and \(\left\{b_1, b_2, \ldots, b_n\right\}\) such that \[ \sum_{i=1}^n a_i^r = \sum_{i=1}^n b_i^r, \qquad r=1,2, \ldots, k. \] The authors determine the number of solutions of the Tarry-Escott problem of size \(3\) and degree \(2\) over a finite field \(\mathbb{Z}_p\) where \(p>3\) is a prime. Moreover, for a prime \(p>5\), they determine the number of solutions of the related system of equations \[ a_1^r +a_2^r + a_3^r = b_1^r + b_2^r + b_3^r, \qquad r=1,2,5 \] over \(\mathbb{Z}_p\).