Affiliation:
1. Jagiellonian University Kraków Poland and State Higher Vocational School in Tarnow Tarnów Poland
Abstract
Abstract
The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points
(
e
k
,
π
(
e
k
)
)
1
∞
$({e_k},\pi ({e_k}))_1^\infty $
. The elements of the sequence (ek)1
∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1
∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then
e
k
+
1
e
k
=
1
${{{e_k} + 1} \over {{e_k}}} = 1$
. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.
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