On the Markoff Equation

Abstract

In recent work by Bourgain, Gamburd and Sarnak, they prove that almost all Markoff numbers (counted with multiplicity) are highly composite. For any fixed \(\nu \geq 0\), they conclude that a (natural) density-1 subset of Markoff numbers have at least \(\nu\) distinct prime factors.

We present a modified version of their argument to conclude our main theorem that the set of Markoff numbers \(r\) such that \(r\) has at least \(f(r)\) prime factors has (natural) density equal to 1, where \(f\) is the function given by a composition of six natural logarithms.

We also present the work of Zagier on the count of Markoff numbers (with multiplicity) below a given bound and Meiri and Puder's recent result on the transitivity of action of the Markoff group on triples of solutions to the Markoff equation on composite moduli. We restrict ourselves to Meiri and Puder's treatment only for the products of primes \(p \equiv 1 \mod 4\), as necessary for the proof of our main theorem.