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Équations de Fermat de type (5, 5,p)

  • Nicolas Billerey (a1)

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Soient p un nombre premier ≥ 7 et d un entier naturel sans puissances cinquièmes. Nous mettons en œuvre les différentes méthodes modulaires connues pour l'étude de l'équation diophantienne x5+y5 = dzp. Nous montrons en particulier qu'elle n'admet aucune solution propre et non triviale pour p ≥ 7 ou pour une infinité de nombres premiers, dans certains cas où d est de la forme 2α.3β.5γ. Pour d = 3, on énonce un critère permettant de vérifier, notamment, que tel est le cas lorsque p est ≤ 106.

Let p be a prime number ≥ 7 and d be a positive integer fifth power free. We use the known modular methods for the study of the diophantine equation x5+y5 = dzp. We prove that this equation has no non trivial proper solution for p ≥ 7 or for infinitely many prime numbers, in some cases where d is of the form 2α.3β.5γ. For d = 3, we give a criterion which allows us to verify that this holds if p is less than 106.

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References

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Équations de Fermat de type (5, 5,p)

  • Nicolas Billerey (a1)

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