1. On a Conjecture of Petrov and Tolev Related to Chen’s Theorem.
- Author
-
Zhou, Guang-Liang and Cai, Yingchun
- Abstract
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 such that if 1 < c < c 0 , then every sufficiently large natural number N can be represented as N = [ p c ] + [ m c ] ,
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 ≤ 1485 / 1484 = 1.00067 ⋯ , every sufficiently large natural number N can be represented as N = [ p c ] + [ m c ] with m having at most 53 prime factors. In this paper, we show that if 1 < c ≤ 1.0198 , then every sufficiently large natural number N can be written as 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF