Abstract:The prime number graph is the set of points $(n,p_n)$ where $p_n$ denotes the $n^{\rm th}$ prime. Let $L(n)$ be the minimum number of straight line segments needed to cover the first $n$ points in this set.
Let $B(n)$ be the largest number of points $(k,p_k)$ with $k\le n$ covered by a single line.
Recently Sloane conjectured that $L(n) = O(n/\log n)$.
We show that $L(n)=O(n \log \log n / \log n)$ and $B(n)\ge c\log n$ for a constant $c>0$ and all
large $n$. Under RH we show that for large $n$ we have
$B(n)=O(n^{3/4}(\log n)^{1/2})$ and
$ L(n)\ge c' n^{1/4} (\log n) ^{-1/2}$ for some constant
$c'>0.$
From: Patrick Solé [view email]
[v1]
Thu, 21 May 2026 17:21:39 UTC (7 KB)
[v2]
Fri, 22 May 2026 13:38:37 UTC (7 KB)
[v3]
Mon, 1 Jun 2026 18:37:45 UTC (9 KB)
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