The Petit theorem of Fermat is a result of the modular Arithmétique.
It is stated as follows. If has is a positive whole unspecified and p a Prime number, then has p - has is a multiple of p .
See also: Arithmetic modular
The knowledge of the structure and particularly of the group of the units of the ring '' Z ''/'' pZ '', allows a simple demonstration of the theorem. If p is a prime number, the group of the units Z / pZ * is a cyclic Groupe of order p - 1, therefore isomorphous with Z /( p - 1) Z .
A first approach consists in considering φ this isomorphism. The image of φ ( has p-1) is equal to ( p - 1) φ ( has ), corresponding to the neutral element of the group. One deduced that has p-1 is the neutral element of Z / pZ *, i.e. unit classifies it, which finishes the demonstration.
A second approach is the application of the theorem of Lagrange, the order of any element of a finished group is a divider about the group. Consequently, if θ is the order of has , then there exists an entirety μ such as θ.μ = p - 1. The entirety has θ is an element of the class of the unit per definition about an element (cf the paragraph Définitions of the cyclic article Groupe) and thus has p - 1= has θ.μ is also element of the class of the unit.
A direct demonstrationThe preceding demonstration is based on the knowledge of the structure of Z /p Z . Such an approach appears with work of Carl Friedrich Gauss in 1801 in its entitled book Disquisitiones arithmeticae. A demonstration not calling upon such a knowledge is nevertheless possible, it is thus accessible to the mathematicians like Fermat, Leibniz or Euler. It is nevertheless longer and more astute.
Demonstration of EulerThere exists another demonstration, of the same order of difficulty as the preceding one and using the Formule of the binomial theorem. This demonstration corresponds to that of Euler published in 1736 or of Leibnz communicated in a handwritten letter not published.
It uses a Raisonnement by recurrence on the value has . On account of the simplicity, the notations used here are that of Gauss, using congruences. If these notations do not correspond to those of Euler, the reasoning is nevertheless identical.
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