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arXiv · 2107.12123

On some symmetries of the base $ n $ expansion of $ 1/m $ : The Class Number connection

Kalyan Chakraborty·Krishnarjun Krishnamoorthy
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Abstract

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 $.

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Kalyan Chakraborty, Krishnarjun Krishnamoorthy. 2021-07-26. On some symmetries of the base $ n $ expansion of $ 1/m $ : The Class Number connection. https://doi.org/10.2140/pjm.2022.319.39

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