On the Frobenius conjecture for Markoff numbers

Text. A triple (a, b, c) of positive integers is called a Markoff triple if it satisfies the Diophantine equation a(2) + b(2) + c(2) = 3abc. A famous old conjecture says that any Markoff triple (a, b, c) with a <= b <= c is determined uniquely by its largest member c. Let (a, b, c) be a Markoff triple with a <= b <= c. In 2001, Button proved that if c is of the form kp(l), where k is an integer with 1 <= k <= 10(35) and p(l) is a prime power, then c uniquely determines a and b. In this paper, as a complement to the result of Button, we prove that if either 3c - 2 or 3c + 2 is of the form kp(l), where k is an integer with 1 <= k <= 10(10) and p(l) is a prime power, then c uniquely determines a and b. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=6J11b51zdSw. (c) 2013 Elsevier Inc. All rights reserved.