On the representation of numbers by binary biquadratic forms

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.
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