Complete solutions of a family of quartic Thue and index form equations
M Mignotte, A Pethö, R Roth - Mathematics of computation, 1996 - ams.org
M Mignotte, A Pethö, R Roth
Mathematics of computation, 1996•ams.orgContinuing the recent work of the second author, we prove that the diophantine
equation\[f_a (x, y)= x^ 4-ax^ 3 yx^ 2 y^ 2+ axy^ 3+ y^ 4= 1\] for $| a|\ge 3$ has exactly 12
solutions except when $| a|= 4$, when it has 16 solutions. If $\alpha=\alpha (a) $ denotes
one of the zeros of $ f_a (x, 1) $, then for $| a|\ge 4$ we also find all $\gamma\in\Bbb Z
[\alpha] $ with $\Bbb Z [\gamma]=\Bbb Z [\alpha] $. References
equation\[f_a (x, y)= x^ 4-ax^ 3 yx^ 2 y^ 2+ axy^ 3+ y^ 4= 1\] for $| a|\ge 3$ has exactly 12
solutions except when $| a|= 4$, when it has 16 solutions. If $\alpha=\alpha (a) $ denotes
one of the zeros of $ f_a (x, 1) $, then for $| a|\ge 4$ we also find all $\gamma\in\Bbb Z
[\alpha] $ with $\Bbb Z [\gamma]=\Bbb Z [\alpha] $. References
Abstract
Continuing the recent work of the second author, we prove that the diophantine equation for has exactly 12 solutions except when , when it has 16 solutions. If denotes one of the zeros of , then for we also find all with . References
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