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x4 + dx2y2 + y2= z2: Some cases with only trivial solutions—and a solution Euler missed

Published online by Cambridge University Press:  18 May 2009

Ezra Brown
Ezra Brown
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
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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 whether

has nontrivial solutions is now answered for all d, 0≤d ≤100.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 3 , September 1989 , pp. 297 - 307
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

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