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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2107.12123 |
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| _version_ | 1866917590332342272 |
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| author | Chakraborty, Kalyan Krishnamoorthy, Krishnarjun |
| author_facet | Chakraborty, Kalyan Krishnamoorthy, Krishnarjun |
| contents | Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \Q(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $.
Secondly, if $ m\equiv 3\mod 4 $, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain congruence relations involving the class number of $ \Q(\sqrt{-m}) $ modulo certain primes $ p $ which properly divide $ n+1 $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_12123 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On some symmetries of the base $ n $ expansion of $ 1/m $ : The Class Number connection Chakraborty, Kalyan Krishnamoorthy, Krishnarjun Number Theory 11R29, 11A07 Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \Q(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Secondly, if $ m\equiv 3\mod 4 $, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain congruence relations involving the class number of $ \Q(\sqrt{-m}) $ modulo certain primes $ p $ which properly divide $ n+1 $. |
| title | On some symmetries of the base $ n $ expansion of $ 1/m $ : The Class Number connection |
| topic | Number Theory 11R29, 11A07 |
| url | https://arxiv.org/abs/2107.12123 |