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J. Sib. Fed. Univ. Math. Phys., 2019, Volume 12, Issue 3, Pages 331–341 (Mi jsfu765)  

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

Full text: PDF file (126 kB)
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Bibliographic databases:

UDC: 511.52
Received: 16.08.2018
Received in revised form: 18.10.2018
Accepted: 01.04.2019
Language:

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
\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{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000471028500008}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. 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  crossref  mathscinet  zmath  isi  scopus
  • Журнал Сибирского федерального университета. Серия "Математика и физика"
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