2024/08/26 by Cormac O’Sullivan, O'Sullivan, Cormac · 2 citations
Computer Science · Mathematics · #11E16 #11E41 #11Y65 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2408.14405
openalex publication_date 2024/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we study, in greater detail than before, J.H. Conway's topographs for integral binary quadratic forms. These are trees in the plane with regions labeled by integers following a simple pattern. Each topograph can display the values of a single form, or represent an equivalence class of forms. We give a new treatment of reduction of forms to canonical equivalence class representatives by employing topographs and a novel continued fraction for complex numbers. This allows uniform reduction for any positive, negative, square or non-square discriminant. Topograph geometry also provides new class number formulas, and short proofs of results of Gauss relating to sums of three squares. Generalizations of the series of Hurwitz for class numbers give evaluations of certain infinite series, summed over the regions or edges of a topograph.