A new theorem on the prime-counting function

TL;DR

This paper investigates conditions under which the prime-counting function (x) can be expressed in a linear form, establishing existence results for such representations and proposing conjectures related to prime divisibility properties.

Contribution

It provides new theorems on when (n) equals a linear function of n, and introduces conjectures on prime divisibility involving the prime-counting function.

Findings

Existence of integers n with (n)=(n+a)/m for certain a and m

For m>4, there exists n with (mn)=m+n

Proposed conjecture on divisibility of prime sums by m+n

Abstract

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 $a\le\lceil e^{m-1}/(m-1)\rceil$ 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.