A new theorem on the prime-counting function.

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 $$a\leqslant \lceil e^{m-1}/(m-1)\rceil $$ , 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 $$m=1,2,3,\ldots $$ , there is a positive integer n such that $$m+n$$ divides $$p_m+p_n$$ , where $$p_k$$ denotes the k-th prime. [ABSTRACT FROM AUTHOR]