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

TL;DR

This paper investigates the connectedness of special points in Markoff mod p graphs, using Fibonacci sequence properties to establish results for primes with specific 2-adic valuations, including Mersenne primes.

Contribution

It provides explicit connectedness results for special points in Markoff mod p graphs for primes with large 2-adic valuations, especially Mersenne primes.

Findings

Special point (1,1,1) lies in a connected component for certain primes.

Connectedness results depend on the 2-adic valuation of p+1.

Explicit results are obtained for primes where p+1 has large 2-adic valuation.

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.