Connectedness of special points in the Markoff mod $p$ graphs

Abstract

It is conjectured that the Markoff equation $X^2+Y^2+Z^2=3XYZ$ satisfies the special Diophantine property that every mod $p$ solution lifts to an integer solution. Progress toward this conjecture has been made by studying the connectedness of the graphs obtained from the action of the Vieta group on the nonzero mod $p$ solutions to the Markoff equation. In this paper, we use results on Pisano periods of the Fibonacci sequence to obtain explicit results on the connectedness of special points in this graph for primes $p$ where $p+1$ has large $2$-adic valuation. In particular, for Mersenne primes $p \equiv \pm 2 \,(\text{mod}\, 5)$, we show that the special point $(1, 1, 1)$ which is fixed under reduction modulo $p$ lies in a component of this graph which is known to be connected.