Let
p
n
{p_n}
denote the nth prime. The prime number graph is the set of lattice points
(
n
,
p
n
)
(n,{p_n})
,
n
=
1
,
2
,
…
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
2{p_n} > {p_{n - i}} + {p_{n + i}}
for all positive
i
>
n
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
p_n^2 > {p_{n - i}}{p_{n + i}}
for all positive
i
>
n
i > n
, thus verifying a conjecture of Selfridge. We make some new conjectures.