Work

Connectivity of Markoff mod‐p graphs and maximal divisors

Year: 2025

Type: article

Abstract: Markoff mod‐ graphs are conjectured to be connected for all primes . In this paper, we use results of Chen and Bourgain, Gamburd, and Sarnak to confirm the conjecture for all . We also provide a method that quickly verifies connectivity for many primes below this bound. In our study of Markoff mod‐ 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 ‐bound by roughly 140 orders of magnitude as compared with an approach using all divisors.

Source: Proceedings of the London Mathematical Society

DOI: 10.1112/plms.70027

Authors: Jillian Eddy, Elena Fuchs, Matthew Litman, Daniel E. Martin, Nico Tripeny

Institutions: University of California, Davis, University College Dublin, Clemson University, University of Pennsylvania

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,700

Funder: National Science Foundation