Connectivity of Markoff mod-p graphs and maximal divisors

2023/08/15 by Jillian Eddy, Eddy, Jillian, Elena Fuchs +7 · 2 citations
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.07579

openalex publication_date 2023/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Markoff mod-p graphs are conjectured to be connected for all primes p. In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all p > 3.448⋅10³⁹². We also provide a method that quickly verifies connectivity for many primes below this bound. In our study of Markoff mod-p graphs we introduce the notion of maximal divisors of a number. We prove sharp asymptotic and explicit upper bounds on the number of maximal divisors, which ultimately improves the Markoff graph p-bound by roughly 140 orders of magnitude as compared with an approach using all divisors.

Cited by

• Topological properties of generalized Markoff mod p graphs
• Divisibility by p for Markoff-like Surfaces

Related

• An almost linear time algorithm testing whether the Markoff graph modulo p is connected
• A new proof of Chen's theorem for Markoff graphs
• Connectedness of special points in the Markoff mod p graphs
• Bounding Lifts of Markoff Triples mod p
• Topological properties of generalized Markoff mod p graphs