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.
Author(s): Elena Deza
Series: Selected Chapters of Number Theory: Special Numbers, 2
Publisher: World Scientific
Year: 2023
Language: English
Pages: 461
City: Singapore
Contents
Preface
About the Author
Notations
1. Preliminaries
1.1. Divisibility of Integers
Division algorithm
Divisibility
Prime and composite numbers
Greatest common divisor and least common multiple
Euclidean algorithm
Coprime numbers
Exercises
1.2. Modular Arithmetic
Congruence relation
Congruence classes
Fermat’s little theorem and Euler’s theorem
Exercises
1.3. Solution to Congruences
Linear congruences
Chinese remainder theorem
Congruences of order m
Exercises
1.4. Quadratic Residues, Legendre Symbol, and Jacobi Symbol
Legendre symbol
Jacobi symbol
Exercises
1.5. Multiplicative Orders, Primitive Roots, and Indexes
Multiplicative order
Primitive roots modulo n
Indexes
Exercises
1.6. Continued Fractions and Their Applications
Definition and simple properties
Applications of continued fractions
Exercises
References
2. Arithmetic Functions
2.1. Additive and Multiplicative Functions
Exercises
2.2. Floor Function and Its Relatives
Exercises
2.3. Möbius Function
Exercises
2.4. Euler’s Totient Function
Exercises
2.5. Prime Counting Functions
Prime numbers
How to recognize whether a natural number is a prime?
Formulas of primes
Fermat numbers
Mersenne numbers
Prime number theorem
Dirichlet’s characters and L-functions
Proof of the Dirichlet’s theorem for the simplest cases
Exercises
2.6. Divisor Functions
Basic notions
Number of divisor function
Sum of divisors function
Aliquot sum function
Unitary divisor function
Mean values of divisor functions
Exercises
References
3. Perfect Numbers
3.1. History of the Question
3.2. Divisor Function and Perfect Numbers
Definition of perfect numbers and divisor functions
Even perfect numbers and Euclid–Euler theorem
Proof of the Euclid-Euler theorem
Mystery of odd perfect numbers
Exercises
3.3. Properties of Perfect Numbers
Historical tricks in the theory of perfect numbers
Primality of 2n − 1 and properties of perfect numbers
Modular properties
Different representations of perfect numbers
Sum-representations of perfect numbers
Other properties of perfect numbers
Exercises
3.4. Search for Perfect Numbers
Numerical methods for perfect numbers
Calculation of odd perfect numbers
Algebraic assumption methods for perfect numbers and Mersenne primes
Prime divisors of Mersenne numbers
Lucas–Lehmer test
Mersenne numbers and prime numbers records
Exercises
3.5. Perfect Numbers in the Family of Special Numbers
Perfect, abundant, and deficient numbers
Perfect numbers and highly composite numbers
Perfect numbers and practical numbers
Perfect numbers and Ore numbers
Perfect numbers and Mersenne numbers
Perfect numbers and Fermat numbers
Perfect numbers and figurate numbers
Perfect numbers and Pascal’s triangle
Perfect numbers and pernicious numbers
Exercises
3.6. Open Problems
Are there any odd perfect numbers?
Are there infinitely many even perfect number?
Lenstra–Pomerance–Wagstaff and Gillies’ conjectures
New Mersenne conjecture
Other open questions
Exercises
References
4. Amicable Numbers
4.1. History of the Question
4.2. Divisor Function and Amicable Numbers
Definition of perfect numbers and divisor functions
Thābit ibn Qurra theorem
Euler’s version of Thābit’s method
How Euler obtained his rule
Analogue of Thābit’s method
4.3. Properties of Amicable Numbers
Simplest properties of amicable pairs
Prime factors of amicable numbers
Sums of amicable pairs
Twin amicable pairs
Distribution of amicable numbers
Exercises
4.4. Search for Amicable Numbers
Numerical methods for amicable pairs
Algebraic assumption methods for amicable numbers
Analogue of Thābit’s method
Constructive methods for amicable pairs
Exercises
4.5. Amicable Numbers in the Family of Special Numbers
Amicable numbers as deficient and abundant numbers
Amicable numbers and aspiring numbers
Amicable numbers and prime numbers
Amicable numbers and Gaussian integers
Exercises
4.6. Open Problems
Exercises
References
5. Generalizations and Analogue
5.1. History of the Question
5.2. Relatives of Perfect Numbers
Semiperfect numbers
Multiply perfect numbers
Hyperperfect numbers
Almost perfect and quasi-perfect numbers
Superperfect numbers
S-perfect numbers
Unitary perfect numbers
Perfect totient numbers
Exercises
5.3. Relatives of Amicable Numbers
Amicable tuples
Friendly numbers
Multiple amicable numbers
Quasi-amicable and augmented amicable numbers
Exercises
5.4. Sociable Numbers
Basic definitions
Properties of sociable numbers
On the behavior of aliquot sequences
Generalizations of sociable numbers
Exercises
5.5. Search for Numbers Under Consideration
Numerical methods for analogue of perfect numbers
Numerical methods for analogue of amicable numbers
Numerical methods for sociable numbers
Algebraic assumption methods for aliquot k-cycles
Algebraic constructive methods for amicable k-tuples
Exercises
5.6. Open Problems
Existence, finiteness, and infiniteness
Catalan’s conjecture
Other problems
Exercises
References
6. Zoo of Numbers
7. Mini Dictionary
8. Exercises
Problems, Connected with Arithmetic Functions
Problems, Connected with Perfect Numbers
Problems, Connected with Amicable Numbers
Solutions to Problems, Connected with Arithmetic Functions
Solutions to Problems, Connected with Perfect Numbers
Solutions to Problems, Connected with Amicable Numbers
Bibliography
Index
