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The diophantine equation a x 2 + b x y + c y 2 = N , D = b 2 - 4 a c > 0

Keith Matthews — 2002

Journal de théorie des nombres de Bordeaux

We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of a x 2 + b x y + c y 2 = N in relatively prime integers x , y , where N ≠ 0 , gcd ( a , b , c ) = gcd ( a , N ) = 1 et D = b 2 - 4 a c > 0 is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation x 2 - D y 2 = N . As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s for...

On fundamental solutions of binary quadratic form equations

Keith R. Matthews;  John P. Robertson;  Anitha 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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