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The Markoff Equation and Solutions Modulo

Student-Run Research Seminar

Speaker: Colby Brown, UC Davis
Location: 2112 MSB
Start time: Wed, May 17 2023, 12:00PM

The Markoff equation is the Diophantine equation . It's solutions are not so hard to enumerate; they form a tree, rooted at , with edges given by the so-called Vieta involution. Less understood are the solutions to the Markoff equation modulo a prime . Modulo , the solutions form graphs , and the shape of these graphs is the subject of several open questions. It is conjectured that is connected for every prime . This conjecture being true would have strong implications for number-theoretic approaches to answering questions about the Markoff solutions in general. In this talk, we summarize the current state of this conjecture, discuss the current approaches to deciding if is connected for various primes , and preview possible avenues for proving connectivity for the values of for which we do not yet know about the connectivity of .


Pizza at 11:50am