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Abstract In 2021, Chen proved that the size of any connected component of the Markoff mod$$p$$ graph is divisible by$$p$$ . In combination with the work of Bourgain, Gamburd, and Sarnak, Chen’s result resolves a conjecture of Baragar for all but finitely many primes: the Markoff mod$$p$$ graph is connected. In particular, strong approximation for Markoff triples holds for all but finitely many primes. We provide an alternative proof of Chen’s theorem.
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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 ofmaximal divisorsof 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.
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- PAR ID:
- 10573147
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 130
- Issue:
- 2
- ISSN:
- 0024-6115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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