A note on the second supplementary law of rational power residue symbols
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910994249285632 |
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| author | Hirakawa, Yoshinosuke Kashio, Tomokazu Sekigawa, Ryutaro Takada, Naoaki Yamamoto, Shuji |
| author_facet | Hirakawa, Yoshinosuke Kashio, Tomokazu Sekigawa, Ryutaro Takada, Naoaki Yamamoto, Shuji |
| contents | As a natural generalization of the Legendre symbol, the $q$-th power residue symbol $(a/p)_q$ is defined for primes $p$ and $q$ with $p\equiv 1 \bmod q$. In this paper, we generalize the second supplementary law by providing an explicit condition for $(q/p)_q = 1$, when $p$ has a special form $p = \sum_{i=0}^{q-1} m^i n^{q-1-i}$. This condition is expressed in terms of the polylogarithm $\mathrm{Li}_{1-q}(x)$ of negative index. Our proof relies on an argument similar to Lemmermeyer's proof of Euler's conjectures for cubic residue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11666 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the second supplementary law of rational power residue symbols Hirakawa, Yoshinosuke Kashio, Tomokazu Sekigawa, Ryutaro Takada, Naoaki Yamamoto, Shuji Number Theory 11A15, 11G55, 11R11, 11R16, 11R18, 11S80 As a natural generalization of the Legendre symbol, the $q$-th power residue symbol $(a/p)_q$ is defined for primes $p$ and $q$ with $p\equiv 1 \bmod q$. In this paper, we generalize the second supplementary law by providing an explicit condition for $(q/p)_q = 1$, when $p$ has a special form $p = \sum_{i=0}^{q-1} m^i n^{q-1-i}$. This condition is expressed in terms of the polylogarithm $\mathrm{Li}_{1-q}(x)$ of negative index. Our proof relies on an argument similar to Lemmermeyer's proof of Euler's conjectures for cubic residue. |
| title | A note on the second supplementary law of rational power residue symbols |
| topic | Number Theory 11A15, 11G55, 11R11, 11R16, 11R18, 11S80 |
| url | https://arxiv.org/abs/2504.11666 |