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All Diophantine equations ax 2 + by 2 + cz 2 = 1 + dxyz, with a, b, c, d N and a d, b d, c d, having solutions (x, y, z) N 3 are determined. One particular equation of this type 2x 2 + 2y 2 + 3z 2 = 1 + 6xyz appeared recently in connection with the description of the lower part of the approximation spectrum for quaternions.
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