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arXiv · 2308.07579

Connectivity of Markoff mod-p graphs and maximal divisors

Jillian Eddy·Elena Fuchs·Matthew Litman·Daniel Martin·Nico Tripeny
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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 $p > 3.448\cdot10^{392}$. 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 \emph{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.

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Jillian Eddy, Elena Fuchs, Matthew Litman, Daniel Martin, Nico Tripeny. 2023-08-15. Connectivity of Markoff mod-p graphs and maximal divisors. https://arxiv.org/abs/2308.07579

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