EXCEPTIONAL SETS IN WARING’S PROBLEM: TWO SQUARES, TWO CUBES AND TWO SIXTH POWERS

Abstract

Let $R(n)$ denote the number of representations of a large positive integer $n$ as the sum of two squares, two cubes and two sixth powers. In this paper, it is proved that the anticipated asymptotic formula of $R(n)$ fails for at most $O(\left( \log X \right)^{2+\varepsilon})$ positive integers not exceeding $X$. This is an improvement of T. D. Wooley's result which requires $O(\left( \log X \right)^{3+\varepsilon})$.