On a Conjecture of Petrov and Tolev Related to Chen’s Theorem

For any real number y, let [y] be the largest integer not exceeding y. Petrov and Tolev conjectured that there exists a constant \(c_{0}>1\) $c_0>1$ such that if \(1<c<c_{0}\) $1<c<c_0$ , then every sufficiently large natural number N can be represented as \(\begin{aligned} N=[p^{c}]+[m^{c}], \end{aligned}\) $$\begin{array}{c}\hfill N=[p^c]+[m^c],\end{array}$$ where p is a prime and m is a natural number having at most 2 prime factors. And, they proved that when c is close to 1, specifically when \(1<c\le 1485/1484=1.00067\dots ,\) $1<c\le 1485/1484=1.00067\cdots ,$ every sufficiently large natural number N can be represented as \(N=[p^{c}]+[m^{c}]\) $N=[p^c]+[m^c]$ with m having at most 53 prime factors. In this paper, we show that if \(1<c\le 1.0198,\) $1<c\le 1.0198,$ then every sufficiently large natural number N can be written as \(N=[p^{c}]+[m^{c}],\) $N=[p^c]+[m^c],$ where p is a prime and m is a natural number having at most 10 prime factors. This improves the result of Petrov and Tolev.