Geodesic
x4 + dx2y2 + y2= z2: Some cases with only trivial solutions—and a solution Euler missed
Glasgow mathematical journal, Tome 31 (1989) no. 3, pp. 297-307

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Mathematicians have studied the diophantine equation of the title ever since the days of Fermat, Leibniz and Euler. In this paper, we review the history of this problem, present several new classes of values of d for which the equation has only trivial solutions, and find a nontrivial solution for d = 85 (a case Euler missed). With these results, the question of whetherhas nontrivial solutions is now answered for all d, 0≤d ≤100. 
Brown, Ezra. x4 + dx2y2 + y2= z2: Some cases with only trivial solutions—and a solution Euler missed. Glasgow mathematical journal, Tome 31 (1989) no. 3, pp. 297-307. doi: 10.1017/S0017089500007862
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