On the Unicity Conjecture for Markoff Numbers

Abstract

Abstract In 1913 Frobenius conjectured that for any positive integer m , there exists at most one pair of integers ( x, y ) with 0 ≤ x ≤ y ≤ m such that ( x, y, m ) is a solution to the Markoff equation: x 2 + y 2 + m 2 = 3 xym . We show this is true if either m , 3 m — 2 or 3 m + 2 is prime, twice a prime or four times a prime.