Prime and thickened prime components in Apollonian circle packings
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908343426088960 |
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| author | Friedlander, Holley Fuchs, Elena Harris, Piper Hsu, Catherine Rickards, James Sanden, Katherine Schindler, Damaris Stange, Katherine E. |
| author_facet | Friedlander, Holley Fuchs, Elena Harris, Piper Hsu, Catherine Rickards, James Sanden, Katherine Schindler, Damaris Stange, Katherine E. |
| contents | Inspired by a question of Sarnak, we introduce the notion of a prime component in an Apollonian circle packing: a maximal tangency-connected subset having all prime curvatures. We also consider thickened prime components, which are augmented by all circles immediately tangent to the prime component. In both cases, we ask about the curvatures which appear. We consider the residue classes attained by the set of curvatures, the number of circles in such components, the number of distinct integers occurring as curvatures, and the number of prime components in a packing. As part of our investigation, we computed and analysed example components up to around curvature $10^{13}$; software is available. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00177 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prime and thickened prime components in Apollonian circle packings Friedlander, Holley Fuchs, Elena Harris, Piper Hsu, Catherine Rickards, James Sanden, Katherine Schindler, Damaris Stange, Katherine E. Number Theory Primary: 52C26, 11D09, 11N32, Secondary: 11E12, 11N36 Inspired by a question of Sarnak, we introduce the notion of a prime component in an Apollonian circle packing: a maximal tangency-connected subset having all prime curvatures. We also consider thickened prime components, which are augmented by all circles immediately tangent to the prime component. In both cases, we ask about the curvatures which appear. We consider the residue classes attained by the set of curvatures, the number of circles in such components, the number of distinct integers occurring as curvatures, and the number of prime components in a packing. As part of our investigation, we computed and analysed example components up to around curvature $10^{13}$; software is available. |
| title | Prime and thickened prime components in Apollonian circle packings |
| topic | Number Theory Primary: 52C26, 11D09, 11N32, Secondary: 11E12, 11N36 |
| url | https://arxiv.org/abs/2410.00177 |