PERFECT AND AMICABLE NUMBERS

This book contains a detailed presentation on the theory of two classes of special numbers, perfect numbers, and amicable numbers, as well as some of their generalizations. It also gives a large list of their properties, facts and theorems with full proofs. Perfect and amicable numbers, as well as most classes of special numbers, have many interesting properties, including numerous modern and classical applications as well as a long history connected with the names of famous mathematicians.

The theory of perfect and amicable numbers is a part of pure Arithmetic, and in particular a part of Divisibility Theory and the Theory of Arithmetical Functions. Thus, for a perfect number n it holds σ(n) = 2n, where σ is the sum-of-divisors function, while for a pair of amicable numbers (n, m) it holds σ(n) = σ(m) = n + m. This is also an important part of the history of prime numbers, since the main formulas that generate perfect numbers and amicable pairs are dependent 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, alongside many important computer algorithms needed for searching for new large elements of these two famous classes of special numbers.

This book contains a list of open problems and numerous questions related to generalizations of the classical case, which provides a broad perspective on the theory of these two classes of special numbers. Perfect and Amicable Numbers can be useful and interesting to both professional and general audiences.

Contents:

• Preliminaries
• Arithmetic Functions
• Perfect Numbers
• Amicable Numbers
• Generalizations and Analogue
• Zoo of Numbers
• Mini Dictionary
• Exercises

Readership: Teachers and students (especially at the university level) interested in Arithmetic, Number Theory, General Algebra, Cryptography and related fields, as well as a general audience of amateur mathematicians. The book can also be used as source material for individual scientific works by undergraduate and postgraduate students.

Key Features:

• In particular, the author expects 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).