Infinitos números de Carmichael

Abstract

The goal of this work is to show that there are infinitely many Carmichael numbers. Hence, the Carmichael numbers are in some way the worst numbers for testing primality using Fermats Little Theorem. Thus, Fermats Little Theorem can be (and is) used as a good test of non-primality, but it never can be used as a primality test. Our main reference was the paper There are infinitely many Carmichael numbers ([1], W. R. Alford, A. Granville and C. Pomerance) and to fulfill our goal we studied many topics in various areas of Mathematics, such as Mertens asymptotic estimates, group theory and characters, Carmichaels function, Davenports constant, Brun-Titchmarsh inequality (which led us to study the Fouriers theory and the large sieve), Prime Number Theorem in Arithmetic Progression in more general hypotheses and some estimates about the zeros of Dirichlet L-series.