In this paper, we show that 0.969ylogx≤π(x)−π(x−y)≤1.031ylogx, where y=xθ,611<θ≤1 with x large enough. In particular, it follows that pn+1−pn<<p6/11+εn for any ε>0, where pn denotes the nth prime.
This is a collection of some of my lesser known, but nonetheless appealing, problems. Its main focus is on problems concerning the representation of integers as sums or products of integers from a given sequence {an}.