Exceptional set for sums of unlike powers of primes

Abstract

Let [Formula: see text] denote the number of representations of an even integer [Formula: see text], [Formula: see text] as the sum of two squares, one cube and three fourth powers of primes, and by [Formula: see text] we denote the number of even integers [Formula: see text], [Formula: see text] such that the expected asymptotic formula for [Formula: see text] fails to hold. In this paper, it is proved that [Formula: see text] for any [Formula: see text].