Sums of primes and squares of primes in short intervals

TL;DR

This paper proves that almost all integers in certain short intervals can be expressed as sums of a prime and a prime square, or as sums of three prime squares, extending understanding of prime representations in short ranges.

Contribution

The paper establishes new results on the representation of integers as sums involving primes and prime squares in short intervals, with bounds as low as $X^{0.33}$.

Findings

Almost all integers in specified short intervals are sums of a prime and a prime square.

Similar results are proven for sums of three prime squares.

Results extend the range of short intervals where prime representations are typical.

Abstract

Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. 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.