Geodesic
On the representation of numbers by binary biquadratic forms
Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 415-422 
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/
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     author = {V. A. Dem'yanenko},
     title = {On~the representation of numbers by binary biquadratic forms},
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     volume = {9},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1969_9_3_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] V. A. Tartakovskii, “Auflösung der Gleichung $x^2-\rho y^4=1$”, Izv. AN SSSR (VI), XX:3, 4 (1926), 301–324 | Zbl

[2] D. K. Faddeev, “Ob uravnenii $x^4-Ay^4=\pm1$”, Tr. Fiz.-matem. in-ta im. V. A. Steklova, V, 1934, 41–52 | Zbl

[3] D. K. Faddeev, “Ob uravnenii $ax^4-by^4=\sigma$, $\sigma=1,2,4,8$”, Uchenye zapiski Leningr. ped. in-ta im. A. I. Gertsena, 28, 1939, 141–146

[4] V. D. Podsypanin, “Ob uravnenii $ax^4+bx^2y^2-cy^4=1$”, Matem. sb., 18(60) (1946), 105–114 | MR

[5] V. D. Podsypanin, “Ob odnom neopredelennom uravnenii chetvertoi stepeni”, Uchenye zapiski Leningr. ped. in-ta im. A. I. Gertsena, 86, 1949, 195–216

[6] A. I. Vinogradov, V. G. Sprindzhuk, “O predstavlenii chisel binarnymi formami”, Matem. zametki, 3:4 (1968), 369–376 | MR | Zbl

[7] A. Baker, “Linear forms in the logarithms of algebraic numbers”, Mathematika, 13 (1966), 204–216 | MR | Zbl

[8] A. Baker, “Linear forms in the logarithms of algebraic numbers”, Mathematika, 14 (1967), 102–107 | MR | Zbl

[9] V. A. Demyanenko, “Otsenka ostatochnogo chlena v formule Teita”, Matem. zametki, 3:3 (1968), 271–278

[10] E. Gekke, Lektsii po teorii algebraicheskikh chisel, Gostekhizdat, M., 1940