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The Diophantine equations $$y^ 2 = x^ 4\pm 1$$ in quadratic fields. (English) Zbl 0167.04005
The author investigates non-trivial solutions of $$y^2=x^4\pm 1$$ in a quadratic field $$\mathbb Q(\sqrt d)$$, $$d$$ rational. With the $$+$$ sign in the equation, he shows that such solutions exist only if $$d=b^4\pm 12b^2+4$$, $$b$$ rational. Similarly, with the $$-$$ sign, $$d$$ must be $$b^4-12b^2+32b-28$$, $$b$$ again rational.
Reviewer: G. L. Watson
##### MSC:
 11D25 Cubic and quartic Diophantine equations