Geodesic
On a problem related to second-order Diophantine equations
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 38-44

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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. 
@article{IIMI_2019_54_a2,
     author = {A. E. Lipin},
     title = {On a problem related to second-order {Diophantine} equations},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {38--44},
     publisher = {mathdoc},
     volume = {54},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2019_54_a2/}
}
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A. E. Lipin. On a problem related to second-order Diophantine equations. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 38-44. http://geodesic.mathdoc.fr/item/IIMI_2019_54_a2/