On the solvability of the quartic equation \(ax^4+bx^2y^2+dy^4=cz^2\) (Q2704409)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the solvability of the quartic equation \(ax^4+bx^2y^2+dy^4=cz^2\) |
scientific article |
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31 August 2001
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quartic diophantine equations
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0.93617034
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0.91034305
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On the solvability of the quartic equation \(ax^4+bx^2y^2+dy^4=cz^2\) (English)
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The Diophantine equation \(ax^4+bx^2y^2+dy^4=cz^2\) has been studied by many authors: Euler, Legendre, Mordell, \dots The authors generalize a criterion of Euler on this equation. They present several consequences of their criteria. For example they prove that if \(k>2\), \(A\) and \(D\) are rational integers such that \(kD^2-1-D^4=A^2\) then \(x=DA\), \(y=(D^4-1)/2\) and \(z=( (k+2)(D^2-1)^4-(k-2)(D^2+1)^4)/16\) form a solution of \(x^4+kx^2y^2+y^4=z^2\); this is simpler than a previous result of Zheng on the same equation. The proofs are elementary.
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