Number Theory

Multiplicative and Additive with Factorization and Primality Testing

Peter D. Schumer

• Publisher: De Gruyter
• Year: 2025
• eBook ISBN: 9783111579542
• Topic: Mathematics
• Subtopic: Computer Science General
• Index: Mathematics

This is a book for an undergraduate number theory course, senior thesis work, graduate level study, or for those wishing to learn about applications of number theory to data encryption and security. With no abstract algebra background required, it covers congruences, the Euclidean algorithm, linear Diophantine equations, the Chinese Remainder Theorem, Mobius inversion formula, Pythagorean triplets, perfect numbers and amicable pairs, Law of Quadratic Reciprocity, theorems on sums of squares, Farey fractions, periodic continued fractions, best rational approximations, and Pell's equation. Results are applied to factoring and primality testing including those for Mersenne and Fermat primes, probabilistic primality tests, Pollard's rho and p-1 factorization algorithms, and others. Also an introduction to cryptology with a full discussion of the RSA algorithm, discrete logarithms, and digital signatures.

Chapters on analytic number theory including the Riemann zeta function, average orders of the lattice and divisor functions, Chebyshev's theorems, and Bertrand's Postulate. A chapter introduces additive number theory with discussion of Waring's Problem, the pentagonal number theorem for partitions, and Schnirelmann density.

Table of contents

• Title Page
• Copyright
• Contents
• Frontmatter
• Contents
• Chapter 1 Background
• Chapter 2 Congruences and prime factorization
• Chapter 3 Arithmetic functions
• Chapter 4 Primitive roots and quadratic reciprocity
• Chapter 5 Sums of squares
• Chapter 6 Continued fractions and Farey sequences
• Chapter 7 Factoring and primality testing
• Chapter 8 Some applications
• Chapter 9 Introduction to analytic number theory
• Chapter 10 Introduction to additive number theory
• Hints and answers to selected exercises
• Index
• Index