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Soit un corps quadratique imaginaire, soient et ses deux -extensions naturelles (la cyclotomique et la prodiédrale), et soit son 2-corps de classes de Hilbert. Soient le complété en 2 de , ou 1, égale à 1 si et seulement si tout diviseur impair de est congru à , ou 1 le 2-rang de Gal, et ou 2 le 2-rang de Gal On a , et des considérations cohomologiques élémentaires nous donnent d’autres contraintes entre , et , mais nous trouvons 2 obstructions supplémentaires de nature...
Soit un entier . Pour et , nous considérons la suite de Lucas . Nous montrons que, pour n’est ni un carré, ni le double, ni le triple d’un carré, ni six fois un carré pour sauf si et .
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