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arXiv · 2509.21094

Power residues, digit expansions and relative class numbers

Kurt Girstmair
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Abstract

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.

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Kurt Girstmair. 2025-09-25. Power residues, digit expansions and relative class numbers. https://arxiv.org/abs/2509.21094

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