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On the application of Skolem’s p-adic method to the solution of Thue equations. (English) Zbl 0674.10012
The authors show by rather ingenious methods that the only integer solutions of the diophantine equation $(*)\quad x^ 4-4x^ 2y^ 2+y^ 4=-47$ are $$(x,y)=(\pm 2,\pm 3)$$ and $$(\pm 3,\pm 2)$$. This seems to be the first case where a totally real binary quartic has been successfully attempted by methods of algebraic number theory.
The authors use a trick of Ljunggren - working in a number field M of degree 8 over $${\mathbb{Q}}$$- to reduce (*) to a p-adic system to which Skolem’s method is successfully applied with prime $$p=71$$. The numerical hardest job is to find all “exceptable” units in M for which task they give a powerful reduction algorithm [see also the second author and B. M. M. de Weger, ibid. 31, No.1, 99-132 (1989; Zbl 0657.10014)].
Reviewer: B.Richter

##### MSC:
 11D25 Cubic and quartic Diophantine equations
Full Text:
##### References:
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