The Uniqueness of the Prime Markoff Numbers

Given the Diophantine equation a²+b²+c²=3abc, a solution tripleof natural numbers (a, b, c) can be arranged in ascending orderso that abc. Then, given the largest element c, one can askwhether this uniquely determines the triple. This is referredto as the Markoff conjecture. The paper proves that, if c isprime, then there is indeed only one triple that solves theequation with c as the largest element. The proof uses onlystandard algebraic number theory, but it was prompted by geometricconsiderations.