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Connectivity of Markoff mod-p graphs and maximal divisors

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arxiv 2308.07579 v1 pith:75I6IJZH submitted 2023-08-15 math.NT

Jillian Eddy , Elena Fuchs , Matthew Litman , Daniel Martin , Nico Tripeny This is my paper · ORCID
classification math.NT
keywords divisorsmarkoffgraphsmaximalboundconnectivitymod-number
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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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  1. Arithmetic and geometry of Markov polynomials

    math.NT 2025-01 conditional novelty 6.0 of 10

    The authors prove the saturation and log-concavity conjectures for Fibonacci and Pell families of Markov polynomials and propose new conjectures about their coefficients and entropy.

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