To make an introduction to a book about arithmetic it is always difficult, because even most apparently simple assertions in this area of study may hide unsuspected inaccuracies, so one must always approach arithmetic with attention and care; and seriousness, because, in spite of the many games based on numbers, arithmetic is not a game. For this reason, I will avoid to do a naive and enthusiastic apology of arithmetic and also to get into a scholarly dissertation on the nature or the purpose of arithmetic. Instead of this, I will summarize this book, which brings together several articles regarding primes and Fermat pseudoprimes, submitted by the author to the preprint scientific database Research Gate.