Sums of primes and squares of primes in short intervals

Abstract

Let ${\mathbf{H}}_2$ denote the set of even integers $n\not\equiv 1(\mathrm{mod}3)$. We prove that when $H\ge X^{0.33}$, almost all integers $n\in {\mathbf{H}}_2$, $X<n\le X+H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.