THE TARRY-ESCOTT PROBLE, OF DEGREE TWO OVER QUADRATIC FIELDS

摘要

The Tarry-Escott problem is to find distinct multisets of integers with the property that the sums of their k-th powers are equal when k is an integer between 1 and n. An ideal solution of the Tarry-Escott problem is one in which each rnultiset has a + 1 elements. In this paper, we extend the Tarry-Escott problem to quadratic number fields Q(d~(1/2)). We also show that there are infinitely many solutions of the Tarry-Escott problem of degree two over an infinite family of rings Z[d~(1/2)]. At the end, we conjecture that there are infinitely many ideal solutions of degree two over quadratic number fields.