Hardy-Ramanujan Journal |
This paper is concerned with the system of simultaneous diophantine equations ∑6i=1Aki=∑6i=1Bki for k=2,4,6,8,10. Till now only two numerical solutions of the system are known. This paper provides an infinite family of solutions. It is well-known that solutions of the above system lead to ideal solutions of the Tarry-Escott Problem of degree 11, that is, of the system of simultaneous equations, ∑12i=1aki=∑12i=1bki for k=1,2,3,…,11. We use one of the ideal solutions to prove new results on sums of 13th powers. In particular, we prove that every integer can be expressed as a sum or difference of at most 27 thirteenth powers of positive integers.