Integral Apollonian circle packings and prime curvatures

2011/05/25 by Jean Bourgain, Bourgain, Jean
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1105.5127

26 pages

openalex publication_date 2011/05/25 · arxiv created 2012/01/12 · arxiv updated 2012/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.

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