Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians. This book gives a complete presentation of the theory of two classes of special numbers (perfect numbers and amicable numbers) and give much of their properties, facts and theorems with full proofs of them.In the book, a complete detailed description of two classes of special numbers, perfect and amicable numbers, as well as their numerous analogue and generalizations, is given. Perfect and amicable numbers, as well as a majority of classes of special numbers, have a long and rich history connected with the names of many famous mathematicians.This is also an important part of the history of prime numbers, since the main formulas generated perfect and amicable pairs, depends on the good choice of one or several primes of special form.Nowadays, the theory of perfect and amicable numbers contains many interesting mathematical facts and theorems, as well as a lot of important computer algorithms needed for searching for new large elements of these two famous classes of special numbers. The mathematical part of this theory is closely connected with classical Arithmetics and Number Theory. It contains information about divisibility properties of perfect and amicable numbers, structure and properties of their generalizations and analogue (sociable numbers, multiperfect numbers, quasiperfect and quasiamicable numbers, etc.), their connections with other classes of special numbers, etc.Moreover, perfect and amicable numbers are involved in the search for new large primes, and have numerous connections with contemporary Cryptography. For these applications, one should study well-known deterministic and probabilistic primality tests, standard algorithms of integer factorization, the questions and open problems of Computational Complexity Theory.Key FeaturesIn particular, we expect to: 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; collect and study a large list of generalizations and relatives of perfect and amicable numbers (sociable numbers, multiperfect numbers, quasiperfect and quasiamicable numbers, etc).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 book Perfect, Amicable and Sociable Numbers: A Computational Approach, Yan S Y , World Scientific, 1996, but this book really has a computational orientation; there is similar book Figurate Numbers, Deza E, Deza M M, World Scientific, 2012; a last, a similar purpose and a similar structure has the first book of the series, Mersenne numbers and Fermat Numbers, Deza E, World Scientific, 2021).

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