Book contents
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Basic properties of the integers
- 2 Congruences
- 3 Computing with large integers
- 4 Euclid's algorithm
- 5 The distribution of primes
- 6 Finite and discrete probability distributions
- 7 Probabilistic algorithms
- 8 Abelian groups
- 9 Rings
- 10 Probabilistic primality testing
- 11 Finding generators and discrete logarithms in
- 12 Quadratic residues and quadratic reciprocity
- 13 Computational problems related to quadratic residues
- 14 Modules and vector spaces
- 15 Matrices
- 16 Subexponential-time discrete logarithms and factoring
- 17 More rings
- 18 Polynomial arithmetic and applications
- 19 Linearly generated sequences and applications
- 20 Finite fields
- 21 Algorithms for finite fields
- 22 Deterministic primality testing
- Appendix: Some useful facts
- Bibliography
- Index of notation
- Index
2 - Congruences
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Preliminaries
- 1 Basic properties of the integers
- 2 Congruences
- 3 Computing with large integers
- 4 Euclid's algorithm
- 5 The distribution of primes
- 6 Finite and discrete probability distributions
- 7 Probabilistic algorithms
- 8 Abelian groups
- 9 Rings
- 10 Probabilistic primality testing
- 11 Finding generators and discrete logarithms in
- 12 Quadratic residues and quadratic reciprocity
- 13 Computational problems related to quadratic residues
- 14 Modules and vector spaces
- 15 Matrices
- 16 Subexponential-time discrete logarithms and factoring
- 17 More rings
- 18 Polynomial arithmetic and applications
- 19 Linearly generated sequences and applications
- 20 Finite fields
- 21 Algorithms for finite fields
- 22 Deterministic primality testing
- Appendix: Some useful facts
- Bibliography
- Index of notation
- Index
Summary
This chapter introduces the basic properties of congruences modulo n, along with the related notion of congruence classes modulo n. Other items discussed include the Chinese remainder theorem, Euler's phi function, arithmetic functions and Möbius inversion, and Fermat's little theorem.
Definitions and basic properties
For positive integer n, and for a, b ∈ ℤ, we say that a is congruent tobmodulon if n | (a - b), and we write a ≡ b (mod n). If n ∣ (a - b), then we write a ≢ b (mod n). The relation a ≡ b (mod n) is called a congruence relation, or simply, a congruence. The number n appearing in such congruences is called the modulus of the congruence. This usage of the “mod” notation as part of a congruence is not to be confused with the “mod” operation introduced in §1.1.
A simple observation is that a ≡ b (mod n) if and only if there exists an integer c such that a = b + cn. From this, and Theorem 1.4, the following is immediate:
Theorem 2.1.Let n be a positive integer. For every integer a, there exists a unique integer b such that a ≡ b (mod n) and 0 ≤ b < n, namely, b ≔ a mod n.
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- Publisher: Cambridge University PressPrint publication year: 2005