Markoff numbers, principal ideals and continued fraction expansions

Given any solution triple of natural numbers to the Markoff equation a(2) + b(2) + c(2) = 3abc, an old problem asks whether the largest number determines the triple uniquely. We show this to be true in a range of cases by considering the factorisation of ideals in certain quadratic number fields, but also exhibit a counterexample for this approach when the question is widened to other numbers. (C) 2001 Academic Press.