Visualizing the Sum-Product Conjecture
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929761036533760 |
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| author | O'Bryant, Kevin |
| author_facet | O'Bryant, Kevin |
| contents | Let $SPP(n)$ be the set $\left\{\big(|A+A|,|A A|\big) : A\subseteq {\mathbb N}, |A|=n\right\}$ of sum-product pairs, where $A+A$ is the sumset $\{a+b : a,b\in A\}$ and $A A$ is the product set $\{ab:a,b\in A\}$. We construct a dataset consisting of 1162868 sets whose sum-product pairs are at least $84\%$ of $SPP(n)$ for each $n\le 32$. Notably, we do **not** see evidence in favor of Erdős's Sum-Product Conjecture in our dataset. For $n\le 6$, we prove the exact value of $SPP(n)$. We include a number of conjectures, open problems, and observations motivated by this dataset, a large number of color visualizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08139 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Visualizing the Sum-Product Conjecture O'Bryant, Kevin Number Theory Combinatorics 11B13 (Primary) 05B10, 11Y99 (Secondary) Let $SPP(n)$ be the set $\left\{\big(|A+A|,|A A|\big) : A\subseteq {\mathbb N}, |A|=n\right\}$ of sum-product pairs, where $A+A$ is the sumset $\{a+b : a,b\in A\}$ and $A A$ is the product set $\{ab:a,b\in A\}$. We construct a dataset consisting of 1162868 sets whose sum-product pairs are at least $84\%$ of $SPP(n)$ for each $n\le 32$. Notably, we do **not** see evidence in favor of Erdős's Sum-Product Conjecture in our dataset. For $n\le 6$, we prove the exact value of $SPP(n)$. We include a number of conjectures, open problems, and observations motivated by this dataset, a large number of color visualizations. |
| title | Visualizing the Sum-Product Conjecture |
| topic | Number Theory Combinatorics 11B13 (Primary) 05B10, 11Y99 (Secondary) |
| url | https://arxiv.org/abs/2411.08139 |