This book contains a complete detailed description of two classes of special numbers closely related to classical problems of the Theory of Primes. There is also extensive discussions of applied issues related to Cryptography. In Mathematics, a Mersenne number (named after Marin Mersenne, who studied them in the early 17-th century) is a number of the form Mn = 2n – 1 for positive integer n. In Mathematics, a Fermat number (named after Pierre de Fermat who first studied them) is a positive integer of the form Fn = 2k+ 1, k=2n, where n is a non-negative integer. Mersenne and Fermat numbers have many other interesting properties. Long and rich history, many arithmetic connections (with perfect numbers, with construction of regular polygons etc.), numerous modern applications, long list of open problems allow us to provide a broad perspective of the Theory of these two classes of special numbers, that can be useful and interesting for both professionals and the general audience. Key Features • The main purpose of the book is to give a complete presentation of the Theory of two classes of special numbers (Mersenne numbers and Fermat numbers) and to give much of their properties, facts and theorems with full proofs of them • In particular, we expect: find and organize much of scattered material; present updated material with all details in clear and unified way; consider all ranges of well-known and hidden connections of a given set number with different mathematical problems; draw up a system of multilevel tasks • Thus, each reader will be able to find the subject of interest at the available level. There are no such books on the Theory of these two classes of Special Numbers (there is a similar book Figurate numbers, Deza E, Deza M M, World Scientific, 2012)

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