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This article is cited in 1 scientific paper (total in 1 paper)
An elementary algorithm for solving a diophantine equation of degree four with Runge's condition
Nikolai N. Osipov, Maria I. Medvedeva Institute of Space and Information Technology, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041, Russia
Abstract:
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:
diophantine equations, elementary version of Runge's method.
DOI:
https://doi.org/10.17516/1997-1397-2019-12-3-331-341
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UDC:
511.52 Received: 16.08.2018 Received in revised form: 18.10.2018 Accepted: 01.04.2019
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Citation:
Nikolai N. Osipov, Maria I. Medvedeva, “An elementary algorithm for solving a diophantine equation of degree four with Runge's condition”, J. Sib. Fed. Univ. Math. Phys., 12:3 (2019), 331–341
Citation in format AMSBIB
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\by Nikolai~N.~Osipov, Maria~I.~Medvedeva
\paper An elementary algorithm for solving a diophantine equation of degree four with Runge's condition
\jour J. Sib. Fed. Univ. Math. Phys.
\yr 2019
\vol 12
\issue 3
\pages 331--341
\mathnet{http://mi.mathnet.ru/jsfu765}
\crossref{https://doi.org/10.17516/1997-1397-2019-12-3-331-341}
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This publication is cited in the following articles:
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Osipov N.N., Dalinkevich S.D., “An Algorithm For Solving a Quartic Diophantine Equation Satisfying Runge'S Condition”, Computer Algebra in Scientific Computing (Casc 2019), Lecture Notes in Computer Science, 11661, eds. England M., Koepf W., Sadykov T., Seiler W., Vorozhtsov E., Springer International Publishing Ag, 2019, 377–392
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