Lines in the prime number graph
Carl Pomerance, Patrick Sol\'e

TL;DR
This paper investigates the geometric properties of the prime number graph, establishing bounds on the number of line segments needed to cover points and the maximum points on a single line, with results depending on the Riemann Hypothesis.
Contribution
It provides new bounds on the covering line segments and collinear points in the prime number graph, improving previous conjectures and conditional results under RH.
Findings
L(n)=O(n log log n / \u0000 log n)
B(n) c log n for large n
Under RH, B(n)=O(n^{3/4}(\u0000 log n)^{1/2}) and L(n) c' n^{1/4} ( log n)^{-1/2}
Abstract
The prime number graph is the set of points where denotes the prime. Let be the minimum number of straight line segments needed to cover the first points in this set. Let be the largest number of points with covered by a single line. Recently Sloane conjectured that . We show that and for a constant and all large . Under RH we show that for large we have and for some constant
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