On the set
ZW Sun, L Zhao�- arXiv preprint arXiv:2004.01080, 2020 - arxiv.org
ZW Sun, L Zhao
arXiv preprint arXiv:2004.01080, 2020•arxiv.orgAn open conjecture of Z.-W. Sun states that for any integer $ n> 1$ there is a positive integer
$ k\le n $ such that $\pi (kn) $ is prime, where $\pi (x) $ denotes the number of primes not
exceeding $ x $. In this paper, we show that for any positive integer $ n $ the set $\{\pi
(kn):\k= 1, 2, 3,\ldots\} $ contains infinitely many $ P_2 $-numbers which are products of at
most two primes. We also prove that under the Bateman--Horn conjecture the set $\{\pi
(4k):\k= 1, 2, 3,\ldots\} $ contains infinitely many primes.
$ k\le n $ such that $\pi (kn) $ is prime, where $\pi (x) $ denotes the number of primes not
exceeding $ x $. In this paper, we show that for any positive integer $ n $ the set $\{\pi
(kn):\k= 1, 2, 3,\ldots\} $ contains infinitely many $ P_2 $-numbers which are products of at
most two primes. We also prove that under the Bateman--Horn conjecture the set $\{\pi
(4k):\k= 1, 2, 3,\ldots\} $ contains infinitely many primes.
An open conjecture of Z.-W. Sun states that for any integer there is a positive integer such that is prime, where denotes the number of primes not exceeding . In this paper, we show that for any positive integer the set contains infinitely many -numbers which are products of at most two primes. We also prove that under the Bateman--Horn conjecture the set contains infinitely many primes.
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