The diophantine equation ax 2 +bxy+cy 2 =N, D=b 2 -4ac>0
Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 257-270.

Nous revisitons un algorithme dû à Lagrange, basé sur le développement en fraction continue, pour résoudre l’équation ax 2 +bxy+cy 2 =N en les entiers x,y premiers entre eux, où N0, pgcd(a,b,c)=pgcd(a,N)=1etD=b 2 -4ac>0 n’est pas un carré.

We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of ax 2 +bxy+cy 2 =N in relatively prime integers x,y, where N0, gcd(a,b,c)=gcd(a,N)=1etD=b 2 -4ac>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 -Dy 2 =N. As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s for the necessity of the existence of a solution. Lagrange did not discuss an exceptional case which can arise when D=5. This was done by M. Pavone in 1986, when N=±μ, where μ=min (x,y)(0,0) ax 2 +bxy+cy 2 . We only need the special case μ=1 of his result and give a self-contained proof, using our unimodular matrix approach.

@article{JTNB_2002__14_1_257_0,
     author = {Matthews, Keith},
     title = {The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {257--270},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {1},
     year = {2002},
     doi = {10.5802/jtnb.358},
     zbl = {1018.11013},
     mrnumber = {1926002},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.358/}
}
Keith Matthews. The diophantine equation $ax^2+bxy+cy^2=N$, $D=b^2-4ac>0$. Journal de Théorie des Nombres de Bordeaux, Tome 14 (2002) no. 1, pp. 257-270. doi : 10.5802/jtnb.358. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.358/

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