Markoff Numbers, Principal Ideals and Continued Fraction Expansions Original Research Article

Given any solution triple of natural numbers to the Markoff equation a2+b2+c2=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.