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.
Received: 16.08.2018 Received in revised form: 18.10.2018 Accepted: 01.04.2019
Bibliographic databases:
Document Type:
Article
UDC:
511.52
Language: English
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
\Bibitem{OsiMed19}
\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}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000471028500008}
Linking options:
https://www.mathnet.ru/eng/jsfu765
https://www.mathnet.ru/eng/jsfu/v12/i3/p331
This publication is cited in the following 2 articles:
N. N. Osipov, A. A. Kytmanov, “An algorithm for solving a family of fourth-degree Diophantine equations that satisfy Runge's condition”, Program. Comput. Softw., 47:1 (2021), 29–33
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
Citation in format AMSBIB
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