Power residues, digit expansions and relative class numbers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912604802252800 |
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| author | Girstmair, Kurt |
| author_facet | Girstmair, Kurt |
| contents | This is a survey of a connection between the distribution of certain power residues modulo $p$, $p$ a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields a number of results about the distribution of quadratic residues, for instance, the well-known fact that the interval $[0,p/2]$ contains more quadratic residues than nonresidues. This class number formula is also responsible for some properties of the digit expansions of numbers $m/p$, $p\NDIV m$. In a certain sense the results based on Dirichlet's formula can be extended to sixth power residues, where geometry plays an important role. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_21094 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Power residues, digit expansions and relative class numbers Girstmair, Kurt Number Theory This is a survey of a connection between the distribution of certain power residues modulo $p$, $p$ a prime, and relative class numbers. The focus lies on quadratic residues and sixth power residues. Dirichlet's class number formula yields a number of results about the distribution of quadratic residues, for instance, the well-known fact that the interval $[0,p/2]$ contains more quadratic residues than nonresidues. This class number formula is also responsible for some properties of the digit expansions of numbers $m/p$, $p\NDIV m$. In a certain sense the results based on Dirichlet's formula can be extended to sixth power residues, where geometry plays an important role. |
| title | Power residues, digit expansions and relative class numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.21094 |