Problems on combinatorial properties of primes [article]

Zhi-Wei Sun
2016 arXiv   pre-print
For x>0 let π(x) be the number of primes not exceeding x. The asymptotic behaviors of the prime-counting function π(x) and the n-th prime p_n have been studied intensively in analytic number theory. Surprisingly, we find that π(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
more » ... any integer n>1 one of the n numbers π(n),π(2n),...,π(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∈{p(n): n=1,2,3,...}.
arXiv:1402.6641v12 fatcat:m5o5wxr23jg5vhbwmtgt5q55vm