Perfect, amicable and sociable numbers: a computational approach

1. Concepts and history. 1.1. Basic concepts. 1.2. The history of amicable numbers. 1.3. Brief history of perfect and sociable numbers -- 2. Mathematical tools. 2.1. Computational complexity theory. 2.2. Primality testing. 2.3. Integer factorization. 2.4. Diophantine equations -- 3. Exhaustive numerical methods. 3.1. Numerical methods for perfect numbers. 3.2. Computing different types of perfect numbers. 3.3. Computing odd perfect numbers. 3.4. Numerical methods for amicable numbers. 3.5. te Riele's seminumerical method for amicable pairs. 3.6. Computing reduced amicable pairs. 3.7. Numerical methods for aliquot k-cycles -- 4. Algebraic assumption methods. 4.1. Thabit's algebraic assumption method. 4.2. Euler's version of Thabit's method. 4.3. Analogue of Thabit's method. 4.4. Some conjectures on amicable pairs. 4.5. Algebraic rules for aliquot k-cycles -- 5. Algebraic constructive methods. 5.1. A special algebraic constructive method. 5.2. A general algebraic constructive method. 5.3. Two other more general constructive methods. 5.4. Algebraic methods for amicable k-tuplets. 5.5. A practical application in cryptography -- 6. Conclusions and open problems. 6.1. Summary. 6.2. Some open problems

This book is about perfect, amicable and sociable numbers, with an emphasis on amicable numbers, from both a mathematical and particularly a computational point of view. Perfect and amicable numbers have been studied since antiquity, nevertheless, many problems still remain. The book introduces the basic concepts and results of perfect, amicable and sociable numbers and reviews the long history of the search for these numbers. It examines various methods, both numerical and algebraic, of generating these numbers, and also includes a set of important and interesting open problems in the area. The book is self-contained, and accessible to researchers, students, and even amateurs in mathematics and computing science. The only prerequisites are some familiarity with high-school algebra and basic computing techniques