On the Frobenius conjecture for Markoff numbers

TextA triple (a,b,c)(a,b,c) of positive integers is called a Markoff triple   if it satisfies the Diophantine equation a2+b2+c2=3abca2+b2+c2=3abc. A famous old conjecture says that any Markoff triple (a,b,c)(a,b,c) with a⩽b⩽ca⩽b⩽c is determined uniquely by its largest member c  . Let (a,b,c)(a,b,c) be a Markoff triple with a⩽b⩽ca⩽b⩽c. In 2001, Button proved that if c   is of the form kpℓkpℓ, where k   is an integer with 1⩽k⩽10351⩽k⩽1035 and pℓpℓ 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−23c−2 or 3c+23c+2 is of the form kpℓkpℓ, where k   is an integer with 1⩽k⩽10101⩽k⩽1010 and pℓpℓ is a prime power, then c uniquely determines a and b.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=6J11b51zdSw.