Geodesic
On fundamental solutions of binary quadratic form equations
Keith R. Matthews 1   ; John P. Robertson 2   ; Anitha Srinivasan 3
1 Department of Mathematics University of Queensland Brisbane 4072, Australia and Centre for Mathematics and its Applications Australian National University Canberra, ACT 0200, Australia
2 Actuarial and Economic Services Division National Council on Compensation Insurance Boca Raton, FL 33487, U.S.A.
3 Department of Mathematics Saint Louis University – Madrid campus Avenida del Valle 34 28003 Madrid, Spain
Acta Arithmetica, Tome 169 (2015) no. 3, pp. 291-299 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa169-3-4
Keywords: suitable modification upper bound estimates stolt fundamental integer solutions diophantine equation buv where positive nonsquare characterize fundamental solutions corollary get corresponding result equation dv where positive nonsquare which upper bound estimates obtained nagell chebyshev

Keith R. Matthews  1   ; John P. Robertson  2   ; Anitha Srinivasan  3

1 Department of Mathematics University of Queensland Brisbane 4072, Australia and Centre for Mathematics and its Applications Australian National University Canberra, ACT 0200, Australia
2 Actuarial and Economic Services Division National Council on Compensation Insurance Boca Raton, FL 33487, U.S.A.
3 Department of Mathematics Saint Louis University – Madrid campus Avenida del Valle 34 28003 Madrid, Spain 
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     title = {On fundamental solutions of binary quadratic form equations},
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Keith R. Matthews; John P. Robertson; Anitha Srinivasan. On fundamental solutions of binary quadratic form equations. Acta Arithmetica, Tome 169 (2015) no. 3, pp. 291-299. doi: 10.4064/aa169-3-4

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