Geodesic
An elementary algorithm for solving a diophantine equation of degree four with Runge's condition
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 3, pp. 331-341

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We propose an elementary algorithm for solving a diophantine equation \begin{equation*} (p(x,y)+a_1x+b_1y)(p(x,y)+a_2x+b_2y)-dp(x,y)-a_3x-b_3y-c=0 \tag{*} \end{equation*} of degree four, where $p(x,y)$ denotes an irreducible quadratic form of positive discriminant and $(a_1,b_1) \neq (a_2,b_2)$. The last condition guarantees that the equation $(*)$ can be solved using the well known Runge's method, but we prefer to avoid the use of any power series that leads to upper bounds for solutions useless for a computer implementation.
Keywords: elementary version of Runge's method.
Mots-clés : diophantine equations 
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     title = {An elementary algorithm for solving a diophantine equation of degree four with {Runge's} condition},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
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Nikolai N. Osipov; Maria I. Medvedeva. An elementary algorithm for solving a diophantine equation of degree four with Runge's condition. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 3, pp. 331-341. http://geodesic.mathdoc.fr/item/JSFU_2019_12_3_a7/