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Random Thue and Fermat equations

Rainer Dietmann; Oscar Marmon — 2015

Acta Arithmetica

We consider Thue equations of the form $a x^{k} + b y^{k} = 1$ , and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $a x^{k} + b y^{k} + c z^{k} = 0$ of degree at least six.

Pairs of cubic forms in many variables

Rainer Dietmann; Trevor D. Wooley — 2003

Acta Arithmetica

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