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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.22752 |
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| _version_ | 1866918518516088832 |
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| author | Pomerance, Carl Solé, Patrick |
| author_facet | Pomerance, Carl Solé, Patrick |
| contents | 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
$c'>0.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22752 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lines in the prime number graph Pomerance, Carl Solé, Patrick Number Theory 11N05 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 $c'>0.$ |
| title | Lines in the prime number graph |
| topic | Number Theory 11N05 |
| url | https://arxiv.org/abs/2605.22752 |