Sums of One Prime Power and Four Prime Cubes in Short Intervals

Abstract

Let k ⩾ 1 be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers n = p 1 k + p 2 3 + p 3 3 + p 4 3 + p 5 3 in short intervals, where p 1 , p 2 , p 3 , p 4 , p 5 are prime numbers.