(p-1) ≡-1 (mod p)


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Exercice 12 Soit E = Rn[X] l'espace vectoriel des polynômes de degré ≤ n et f : E → E définie par: f(P) = P+(1−X)P Montrer que f est une application 

  • Pourquoi 1 est un nombre premier ?

    Selon cette définition, les nombres 0 et 1 ne sont donc ni premiers ni composés : 1 n'est pas premier car il n'a qu'un seul diviseur entier positif et 0 non plus car il est divisible par tous les entiers positifs.

  • Le moyen brutal de déterminer si un nombre n est premier est de le diviser par tous les nombres impairs inférieurs à √n (si n est composé, un de ses diviseurs est inférieur à √n) : cette manière de procéder n'est pas polynomiale, mais exponentielle parce que si n est de taille t, il y a environ 10t/2 nombres inférieurs
En théorie des nombres, l'algorithme p – 1 de Pollard est un algorithme de décomposition en produit de facteurs premiers, conçu par John M. Pollard en 1974.Autres questions
  • What does mod p mean?

    Definition. ( i) The “mod p” numbers are all the remainders: {0,1,2,,p ? 1} when a natural number is divided by p.
  • What is ZP * in maths?

    Z/pZ, the cyclic group of integers modulo p.
  • What are the elements of ZP?

    Since Zp is an “integral domain” we know that x = y or x = ?y mod p. Hence, elements in Zp have either zero square roots or two square roots. If a is the square root of x then ?a is also a square root of x modulo p.
  • Usually the notation Zp or Z/pZ mean the integers modulo p, that is {0,…,p?1} where you add and multiply as usually and then take the reminder modulo p. The notation of Z×p is for those numbers which have a multiplicative inverse modulo p, namely all n such that exists m such that n?m is 1 modulo p.
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