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Sur les 𝐙 2 -extensions d’un corps quadratique imaginaire

Georges Gras (1983)

Annales de l'institut Fourier

Soit k = Q ( - m ) un corps quadratique imaginaire, soient k et F ses deux Z 2 -extensions naturelles (la cyclotomique et la prodiédrale), et soit k ˇ son 2-corps de classes de Hilbert. Soient 𝒫 le complété en 2 de k , ρ = 0 ou 1, égale à 1 si et seulement si tout diviseur impair de m est congru à ± 1 mod 8 , χ = 0 ou 1 le 2-rang de Gal ( k F / k ) , et t = 0 , 1 ou 2 le 2-rang de Gal k ˇ F k ˇ / k ) . On a χ ρ , et des considérations cohomologiques élémentaires nous donnent d’autres contraintes entre 𝒫 , χ et t , mais nous trouvons 2 obstructions supplémentaires de nature...

Sur les carrés dans certaines suites de Lucas

Maurice Mignotte, Attila Pethö (1993)

Journal de théorie des nombres de Bordeaux

Soit a un entier 3 . Pour α = ( a + a 2 - 4 ) / 2 et β = ( a - a 2 - 4 ) / 2 , nous considérons la suite de Lucas 𝑢 𝑛 = ( α 𝑛 - β 𝑛 ) / ( α - β ) . Nous montrons que, pour a 4 , 𝑢 𝑛 n’est ni un carré, ni le double, ni le triple d’un carré, ni six fois un carré pour n > 3 sauf si a = 338 et n = 4 .

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