Problems on combinatorial properties of primes

Abstract

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find that $\pi(x)$ and $p_n$ have many combinatorial properties which should not be ignored. In this paper we pose 60 open problems on combinatorial properties of primes (including connections between primes and partition functions) for further research. For example, we conjecture that for any integer $n>1$ one of the $n$ numbers $\pi(n),\pi(2n),...,\pi(n^2)$ is prime; we also conjecture that for any integer $n>6$ there exists a prime $p<n$ such that $pn$ is a primitive root modulo $p_n$. One of our conjectures involving the partition function $p(n)$ states that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g\in\{p(n):\ n=1,2,3,...\}$.