Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 21 (2026), 157 -- 176
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POWER RESIDUES, DIGIT EXPANSIONS, AND RELATIVE CLASS NUMBERS

Kurt Girstmair

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 ∤ 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.

2020 Mathematics Subject Classification: 11A15; 11A63; 11R29.
Keywords: Power residues; digit expansions; relative class numbers; sign vectors.

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Kurt Girstmair, ORCID: https://orcid.org/0000-0003-3105-5111
Institut für Mathematik, Universität Innsbruck,
Technikerstr. 13/7, A-6020 Innsbruck, Austria.
e-mail: kurt.girstmair@uibk.ac.at





Received: September 29, 2025; Accepted: April 22, 2026
Published electronically: April 23, 2026