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Izv. IMI UdGU, 2019, Volume 54, Pages 38–44 (Mi iimi380)  

On a problem related to second-order Diophantine equations

A. E. Lipin

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620219, Russia

Abstract: The article considers the problem set by V. N. Ushakov of finding triangles with integer lengths of sides $a$, $b$, $c$, satisfying the relations $a^2=b^2+c^2+k$ and $\dfrac{a}{c}=\dfrac{3}{2}$, where $k$ is a nonzero integer. We give a necessary and sufficient condition for the number $k$ under which such triangles exist. The proof is constructive and allows, in the case of satisfying the criterion, to indicate an infinite number of triples $(a,b,c)$ with the given property.

Keywords: systems of diophantine equations, recurrence relations, Fibonacci and Lucas numbers.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00221_
The studies was funded by RFBR, project number 18–0100221.


DOI: https://doi.org/10.20537/2226-3594-2019-54-03

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Bibliographic databases:

UDC: 511.176
MSC: 11B39
Received: 26.09.2019

Citation: A. E. Lipin, “On a problem related to second-order Diophantine equations”, Izv. IMI UdGU, 54 (2019), 38–44

Citation in format AMSBIB
\Bibitem{Lip19}
\by A.~E.~Lipin
\paper On a problem related to second-order Diophantine equations
\jour Izv. IMI UdGU
\yr 2019
\vol 54
\pages 38--44
\mathnet{http://mi.mathnet.ru/iimi380}
\crossref{https://doi.org/10.20537/2226-3594-2019-54-03}
\elib{https://elibrary.ru/item.asp?id=41435139}


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