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Industrie des polynômes valeurs entières diviseurs nombres premiers

4 pages
Industrie des polynômes à valeurs entières : diviseurs, nombres premiers, « petit » théorème de Fermat, records I – L'USINE 2 II – LES DIVISEURS DES POLYNÔMES 3 Une bonne question 3 Quels diviseurs ? 3 On revient à P (x) = x 5 – x 4 Et la démonstration du théorème A ? 5 III – LE « PETIT » THÉORÈME DE FERMAT 6 En préambule : le triangle arithmétique de Pascal 6 Le « petit » théorème de Fermat 6 IV – LES POLYNÔMES QUI « DONNENT » DES NOMBRES PREMIERS 9 1772 : Euler 9 Comment fait-on pour trouver des records ? 9 Les « diviseurs premiers périodiques » 10 Le record de Ruby a-t-il été battu ? 11 1/12

  • divisibilité par pa et par qb

  • triangle arithmétique de pascal

  • temps au temps

  • polynôme

  • formule

  • boutons de réglage

  • sacoche pleine de théorèmes

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292 C.VOLL (APPENDIX BY A. BEAUVILLE)GAFA Appendix: Lineson Pfaffian Hypersurfaces by A. Beauville The aim of this appendix is to prove that a general pfaffian hypersurface n of degreer >2nŠ3 inPcontains no lines (Proposition 1). By a simple dimension count (see Corollary 4 below), it suffices to show that the variety of lines contained in the universal pfaffian hypersurface (that is, the hyper-surface of degenerate forms in the space of all skew-symmetric forms on a given vector space) has the expected dimension. We will deduce this from an explicit description of the pencils of degenerate skew-symmetric forms, which is the content of the proposition below. We work over an algebraically closed “eldk. We will need an elementary lemma: Lemma 4.Given a pencil of skew-symmetric forms on an-dimensional ! " n+1 vector space, there exists a subspace of dimensionwhich is isotropic 2 for all forms of the pencil. Proof. By induction onn, the casesn= 0 andn= 1 being trivial. Let ϕ+be our pencil; we can assume thatϕis degenerate. LetDbe a line contained in the kernel ofϕ, and letDbe its orthogonal with respect ¯ toψ. Thenϕandψinduce skew-symmetric forms¯ϕandψonD /D; ! " nŠ1 by the induction hypothesis there exists a subspace of dimensionin 2 ¯ ⊥ ⊥ D /Dwhich is isotropic for¯ϕandψ. The pull-back of this subspace inD ! " n+1 has dimensionand is isotropic forϕandψ.2 The following result must be well known, but I have not been able to “ nda reference: 1 Proposition 5.LetVbe a vector space of dimension2r, and(ϕt)Pa t pencil ofdegenerateskew-symmetric forms onV. There exists a subspace 1 LVof dimensionr+ 1which is isotropic forϕtfor alltP. Proof. Again we prove the proposition by induction onr, the caser= 1 be-ing trivial. The associated maps Φt:VVform a pencil of singular linear maps. By a classical result in linear algebra (see [G, Chap.XII, Thm.4]),  ∗there exist subspacesKVandLV, with dimK= dimL+ 1, such that Φt(K)Lfor allt; equivalently, there exist subspacesKandLofV, with dimK+ dimL= 2r+ 1, which are orthogonal for eachϕt. Replacing (K, L) by (KL, K+L) we may assumeKL; the pencil (ϕt) restricted toLis singular onK, hence induces a pencil (¯ϕt) onL/K. Put dimK=p, so that dim(L/K) = 2r+ 1Š2p. By the above lemma there is a subspace