Connectivity of Markoff mod‐p graphs and maximal divisors

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.