Geodesic
On~the representation of numbers by binary biquadratic forms
Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 415-422

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In this paper it is proved that if the rank of the equation $ax^4+bx^2y^2+cy^4=kz^2$ over the field $R(1)$ does not exceed unity, and if $k$ is not divisible by any fourth power and is relatively prime to the discriminant, then, provided that $\frac{(b^2-4ac)}{\max\{|a|,|c|\}}$ is sufficiently large relative to $k$, the equation $ax^4+bx^2y^2+cy^4=k$ does not have more than three positive integer solutions. Bibliography: 10 titles. 
@article{SM_1969_9_3_a8,
     author = {V. A. Dem'yanenko},
     title = {On~the representation of numbers by binary biquadratic forms},
     journal = {Sbornik. Mathematics},
     pages = {415--422},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {1969},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1969_9_3_a8/}
}
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V. A. Dem'yanenko. On~the representation of numbers by binary biquadratic forms. Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 415-422. http://geodesic.mathdoc.fr/item/SM_1969_9_3_a8/