科研之友
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摘要
For x > 0, let pi(x) denote the number of primes not exceeding x. For integers a and m > 0, we determine when there is an integer n > 1 with pi(n) = (n+a)/ m. In particular, we show that, for any integers m > 2 and a <= [ e(m-1)/(m - 1,] there is an integer n > 1 with pi(n) = (n+a) / m. Consequently, for any integer m > 4, there is a positive integer n with pi(mn) = m + n. We also pose several conjectures for further research; for example, we conjecture that, for each m = 1, 2, 3, ... , there is a positive integer n such that m n divides p(m) + p(n) where p(k) denotes the k-th prime.
- 出版日期2017-1
- 单位南京大学
