The prime number graph

Abstract:

Let ${p_n}$ denote the nth prime. The prime number graph is the set of lattice points $(n,{p_n})$, $n = 1,2, \ldots$. We show that for every k there are k such points that are collinear. By considering the convex hull of the prime number graph, we show that there are infinitely many n such that $2{p_n} < {p_{n - i}} + {p_{n + i}}$ for all positive $i < n$. By a similar argument, we show that there are infinitely many n for which $p_n^2 > {p_{n - i}}{p_{n + i}}$ for all positive $i < n$, thus verifying a conjecture of Selfridge. We make some new conjectures.