Class Number Growth in Imaginary Quadratic Orders at Gaussian Prime Norm Discriminants

We prove that the class numbers of imaginary quadratic orders at cascade-determined discriminants form a geometric tower: h(−4 × 52k) = 2 × 5k−1 for k ≥ 1, with growth factor N(2+i) = 5—the seed norm. Class numbers at cascade-component conductors are themselves cascade quantities: h(7) = 4 = |Z4|, h(17) = 8 = max(Bit+), h(137) = 68 = 4 × 17. The class number map sends cascade numbers to cascade numbers in a closed, self-reinforcing network.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-31
DOI
https://doi.org/10.5281/zenodo.23843300
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

Class Number Growth in Imaginary Quadratic Orders at Gaussian Prime Norm Discriminants

Robert A. Kenney
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Class Number Growth in Imaginary Quadratic Orders at Gaussian Prime Norm Discriminants

Robert A. Kenney
preprint en

Abstract

We prove that the class numbers of imaginary quadratic orders at cascade-determined discriminants form a geometric tower: h(−4 × 52k) = 2 × 5k−1 for k ≥ 1, with growth factor N(2+i) = 5—the seed norm. Class numbers at cascade-component conductors are themselves cascade quantities: h(7) = 4 = |Z4|, h(17) = 8 = max(Bit+), h(137) = 68 = 4 × 17. The class number map sends cascade numbers to cascade numbers in a closed, self-reinforcing network.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
DOIOpen Access PDFOpenAlex
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Class Number Growth in Imaginary Quadratic Orders at Gaussian Prime Norm Discriminants — Robert A. Kenney · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS