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.