The sequence of prime gaps is graphic
Fuente:
arXiv
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| Format: | Preprint |
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2022
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| author | Erdős, Péter L. Harcos, Gergely Kharel, Shubha R. Maga, Péter Mezei, Tamás R. Toroczkai, Zoltán |
| author_facet | Erdős, Péter L. Harcos, Gergely Kharel, Shubha R. Maga, Péter Mezei, Tamás R. Toroczkai, Zoltán |
| contents | Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann hypothesis. Moreover, an infinite sequence of prime gap graphs can be generated by the so-called degree preserving growth process. This is the first time a naturally occurring infinite sequence of positive integers is identified as graphic. That is, we show the existence of an interesting, and so far unique, infinite combinatorial object. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_00580 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The sequence of prime gaps is graphic Erdős, Péter L. Harcos, Gergely Kharel, Shubha R. Maga, Péter Mezei, Tamás R. Toroczkai, Zoltán Combinatorics Number Theory Primary 05C07, 11N05, Secondary 05C70, 11M26 Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann hypothesis. Moreover, an infinite sequence of prime gap graphs can be generated by the so-called degree preserving growth process. This is the first time a naturally occurring infinite sequence of positive integers is identified as graphic. That is, we show the existence of an interesting, and so far unique, infinite combinatorial object. |
| title | The sequence of prime gaps is graphic |
| topic | Combinatorics Number Theory Primary 05C07, 11N05, Secondary 05C70, 11M26 |
| url | https://arxiv.org/abs/2205.00580 |