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Markoff numbers and ambiguous classes

Anitha Srinivasan — 2009

Journal de Théorie des Nombres de Bordeaux

The Markoff conjecture states that given a positive integer c , there is at most one triple ( a , b , c ) of positive integers with a b c that satisfies the equation a 2 + b 2 + c 2 = 3 a b c . The conjecture is known to be true when c is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant d = 9 c 2 - 4 , every ambiguous form in the principal genus corresponds to a divisor of 3 c - 2 , then the conjecture is true. As a result, we obtain criteria in terms of...

On fundamental solutions of binary quadratic form equations

Keith R. MatthewsJohn P. RobertsonAnitha Srinivasan — 2015

Acta Arithmetica

We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation Au²+Buv+Cv²=N, where A>0, N≠0 and B²-4AC is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation u²-dv²=N, where d is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.

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