Название: Perfect and Amicable Numbers, 1st Edition
Автор: Elena Deza
Издательство: World Scientific Publishing
Серия: Selected Chapters of Number Theory: Special Numbers
Год: 2023
Страниц: 462
Язык: английский
Формат: pdf (true)
Размер: 15.6 MB

Special numbers, among integers, are an important part of number theory, general mathematics, and several applied areas (such as cryptography). While the “names” of some of them (Fermat numbers, Mersenne numbers, Fibonacci numbers, Catalan numbers, Bernoulli numbers, Stirling numbers, etc.) are known to every mathematician, actual information on them is often scattered in the special literature. There are only few books which systematically present their history and properties while giving the main proofs.

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.

Nowadays, the theory of perfect and amicable numbers contains many interesting mathematical facts and theorems as well as a lot of important computer algorithms for searching new large elements of these two famous classes of special numbers.

The mathematical part of this theory is closely connected with classical arithmetic and number theory. It contains much information about divisibility properties of perfect and amicable numbers, structure and properties of their generalizations and analogues (sociable numbers, multiperfect numbers, quasi-perfect and quasi-amicable numbers, etc.), their connections with other classes of special numbers, etc.

Perfect and Amicable Numbers can be useful and interesting to both professional and general audiences.