On fundamental solutions of binary quadratic form equations
Volume 169 / 2015
Acta Arithmetica 169 (2015), 291-299
MSC: Primary 11A55, 11D09, 11D45.
DOI: 10.4064/aa169-3-4
Abstract
We show that, with suitable modification, the upper bound estimates of Stolt for the fundamental integer solutions of the Diophantine equation $Au^2+Buv+Cv^2=N$, where $A>0$, $N\not =0$ and $B^2-4AC$ is positive and nonsquare, in fact characterize the fundamental solutions. As a corollary, we get a corresponding result for the equation $u^2-dv^2=N$, where $d$ is positive and nonsquare, in which case the upper bound estimates were obtained by Nagell and Chebyshev.
