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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-3-4,
author = {Keith R. Matthews and John P. Robertson and Anitha Srinivasan},
title = {On fundamental solutions of binary quadratic form equations},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {291-299},
zbl = {06451621},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-3-4}
}
Keith R. Matthews; John P. Robertson; Anitha Srinivasan. On fundamental solutions of binary quadratic form equations. Acta Arithmetica, Tome 168 (2015) pp. 291-299. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-3-4/