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. Part One of this book, “Sequences of primes and conjectures on them”, brings together thirty-two papers regarding sequences of primes, sequences of squares of primes, sequences of certain types of semiprimes, also few types of pairs, triplets and quadruplets of primes and conjectures on all of these sequences. There are also few papers regarding possible methods to obtain large primes or very large numbers with very few prime factors, some of them based on concatenation, some of them on other arithmetic operations. It is also introduced a new notion: “Smarandache-Coman sequences of primes”, defined as “all sequences of primes obtained from the terms of Smarandache sequences using any arithmetical operation” (for instance, the sequence of primes obtained concatenating to the right with the digit one the terms of Smarandache consecutive numbers sequence). Part Two of this book, “Sequences of Fermat pseudoprimes and conjectures on them”, brings together seventeen papers on sequences of Poulet numbers and Carmichael numbers, i.e. the Fermat pseudoprimes to base 2 and the absolute Fermat pseudoprimes, two classes of numbers that fascinated the author for long time. Among these papers there is a list of thirty-six polynomials and formulas that generate sequences of Fermat pseudoprimes. Part Three of this book, “Prime producing quadratic polynomials”, contains three papers which list some already known such polynomials, that generate more than 20, 30 or even 40 primes in a row, and few such polynomials discovered by the author himself (in a review of records in the field of prime generating polynomials, written by Dress and Landreau, two French mathematicians well known for records in this field, review that can be found on the web address , the author – he says this proudly, of course – is mentioned with 18 prime producing quadratic polynomials). One of the papers proposes seventeen generic formulas that may generate prime-producing quadratic polynomials.