This chapter discusses some of the basic properties of the integers, including the notions of divisibility and primality, unique factorization into primes, greatest common divisors, and least common multiples.

Divisibility and primality

A central concept in number theory is divisibility.

Consider the integers ℤ = {…,−2,−1, 0, 1, 2, …}. For a, b ∈ ℤ, we say that adividesb if az = b for some z ∈ ℤ. If a divides b, we write a | b, and we may say that a is a divisor of b, or that b is a multiple of a, or that b is divisible bya. If a does not divide b, then we write.

We first state some simple facts about divisibility:

Theorem 1.1. For all a, b, c ∈ ℤ, we have

1. (i) a | a, 1 | a, and a | 0;

2. (ii) 0 | a if and only if a = 0;

3. (iii) a | b if and only if − a | b if and only if a | −b;

4. (iv) a | b and a | c implies a | (b + c);

5. (v) a | b and b | c implies a | c.

Proof. These properties can be easily derived from the definition of divisibility, using elementary algebraic properties of the integers. For example, a | a because we can write a · 1 = a; 1 · a because we can write 1 · a = a; a | 0 because we can write a · 0 = 0.

This chapter introduces the basic properties of congruences modulo n, along with the related notion of residue classes modulo n. Other items discussed include the Chinese remainder theorem, Euler's phi function, Euler's theorem, Fermat's little theorem, quadratic residues, and finally, summations over divisors.

Equivalence relations

Before discussing congruences, we review the definition and basic properties of equivalence relations.

Let S be a set. A binary relation ∼ on S is called an equivalence relation if it is

reflexive:a ∼ a for all a ∈ S,

symmetric:a ∼ b implies b ∼ a for all a, b ∈ S, and

transitive:a ∼ b and b ∼ c implies a ∼ c for all a, b, c ∈ S.

If ∼ is an equivalence relation on S, then for a ∈ S one defines its equivalence class as the set {x ∈ S : x ∼ a}.

Theorem 2.1. Let ∼ be an equivalence relation on a set S, and for a ∈ S, let [a] denote its equivalence class. Then for all a, b ∈ S, we have:

1. (i) a ∈ [a];

2. (ii) a ∈ [b] implies [a] = [b].

Proof. (i) follows immediately from reflexivity. For (ii), suppose a ∈ [b], so that a ∼ b by definition. We want to show that [a] = [b]. To this end, consider any x ∈ S.

We establish here some terminology, notation, and simple facts that will be used throughout the text.

Logarithms and exponentials

We write log x for the natural logarithm of x, and logbx for the logarithm of x to the base b.

We write ex for the usual exponential function, where e ≈ 2.71828 is the base of the natural logarithm. We may also write exp[x] instead of ex.

Sets and families

We use standard set-theoretic notation: ∅ denotes the empty set; x ∈ A means that x is an element, or member, of the set A; for two sets A, B, A ⊆ B means that A is a subset of B (with A possibly equal to B), and A ⊊ B means that A is a proper subset of B (i.e., A ⊆ B but A ≠ B). Further, A ∪ B denotes the union of A and B, A ∩ B the intersection of A and B, and A \ B the set of all elements of A that are not in B. If A is a set with a finite number of elements, then we write |A| for its size, or cardinality. We use standard notation for describing sets; for example, if we define the set S ≔ {−2,−1, 0, 1, 2}, then {x2 : x ∈ S} = {0, 1, 4} and {x ∈ S : x is even} = {−2, 0, 2}.

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. My goal in writing this book was to provide an introduction to number theory and algebra, with an emphasis on algorithms and applications, that would be accessible to a broad audience. In particular, I wanted to write a book that would be appropriate for typical students in computer science or mathematics who have some amount of general mathematical experience, but without presuming too much specific mathematical knowledge.

Prerequisites. The mathematical prerequisites are minimal: no particular mathematical concepts beyond what is taught in a typical undergraduate calculus sequence are assumed.

The computer science prerequisites are also quite minimal: it is assumed that the reader is proficient in programming, and has had some exposure to the analysis of algorithms, essentially at the level of an undergraduate course on algorithms and data structures.

Even though it is mathematically quite self contained, the text does presuppose that the reader is comfortable with mathematical formalism and also has some experience in reading and writing mathematical proofs. Readers may have gained such experience in computer science courses such as algorithms, automata or complexity theory, or some type of “discrete mathematics for computer science students” course. They also may have gained such experience in undergraduate mathematics courses, such as abstract or linear algebra.

This chapter introduces the notion of an abelian group. This is an abstraction that models many different algebraic structures, and yet despite the level of generality, a number of very useful results can be easily obtained.

Definitions, basic properties, and examples

Definition 6.1. An abelian group is a set G together with a binary operation * on G such that:

1. (i) for all a, b, c ∈ G, a * (b * c) = (a * b) c (i.e., * is associative);

2. (ii) there exists e ∈ G (called the identity element) such that for all a ∈ G, a * e = a = e * a;

3. (iii) for all a ∈ G there exists a′ ∈ G (called the inverse of a) such that a * a′ = e = a′ * a;

4. (iv) for all a, b ∈ G, a * b = b * a (i.e., * is commutative).

While there is a more general notion of a group, which may be defined simply by dropping property (iv) in Definition 6.1, we shall not need this notion in this text. The restriction to abelian groups helps to simplify the discussion significantly. Because we will only be dealing with abelian groups, we may occasionally simply say “group” instead of “abelian group.”

In this chapter, we review standard asymptotic notation, introduce the formal computational model that we shall use throughout the rest of the text, and discuss basic algorithms for computing with large integers.

Asymptotic notation

We review some standard notation for relating the rate of growth of functions. This notation will be useful in discussing the running times of algorithms, and in a number of other contexts as well.

Let f and g be real-valued functions. We shall assume that each is defined on the set of non-negative integers, or, alternatively, that each is defined on the set of non-negative reals. Actually, as we are only concerned about the behavior of f(x) and g(x) as x → ∞, we only require that f(x) and g(x) are defined for all sufficiently large x (the phrase “for all sufficiently large x” means “for some x0 and all x ≥ x0”). We further assume that g is eventually positive, meaning that g(x) > 0 for all sufficiently large x. Then

• f = O(g) means that |f(x)| ≤ cg(x) for some positive constant c and all sufficiently large x (read, “f is big-O of g”),

• f = Ω(g) means that f(x) ≥ cg(x) for some positive constant c and all sufficiently large x (read, “f is big-Omega of g”),

• f = Θ(g) means that cg(x) ≤ f(x) ≤ dg(x) for some positive constants c and d and all sufficiently large x (read, “f is big-Theta of g”),

• […]

It is sometimes useful to endow our algorithms with the ability to generate random numbers. In fact, we have already seen two examples of how such probabilistic algorithms may be useful:

• at the end of §3.4, we saw how a probabilistic algorithm might be used to build a simple and efficient primality test; however, this test might incorrectly assert that a composite number is prime; in the next chapter, we will see how a small modification to this algorithm will ensure that the probability of making such a mistake is extremely small;

• in §4.5, we saw how a probabilistic algorithm could be used to make Fermat's two squares theorem constructive; in this case, the use of randomization never leads to incorrect results, but the running time of the algorithm was only bounded “in expectation.”

We will see a number of other probabilistic algorithms in this text, and it is high time that we place them on a firm theoretical foundation. To simplify matters, we only consider algorithms that generate random bits. Where such random bits actually come from will not be of great concern to us here. In a practical implementation, one would use a pseudo-random bit generator, which should produce bits that “for all practical purposes” are “as good as random.” While there is a well-developed theory of pseudo-random bit generation (some of which builds on the ideas in §8.9), we will not delve into this here.

This chapter introduces the notion of a ring, more specifically, a commutative ring with unity. While there is a lot of terminology associated with rings, the basic ideas are fairly simple. Intuitively speaking, a ring is an algebraic structure with addition and multiplication operations that behave as one would expect.

Definitions, basic properties, and examples

Definition 7.1. A commutative ring with unity is a set R together with addition and multiplication operations on R, such that:

1. (i) the set R under addition forms an abelian group, and we denote the additive identity by 0R;

2. (ii) multiplication is associative; that is, for all a, b, c ∈ R, we have a(bc) = (ab)c;

3. (iii) multiplication distributes over addition; that is, for all a, b, c ∈ R, we have a (b + c) = ab + ac and (b + c)a = ba + ca;

4. (iv) there exists a multiplicative identity; that is, there exists an element 1R ∈ R, such that 1R · a = a = a · 1Rfor all a ∈ R;

5. (v) multiplication is commutative; that is, for all a, b ∈ R, we have ab = ba.

There are other, more general (and less convenient) types of rings–one can drop properties (iv) and (v), and still have what is called a ring. We shall not, however, be working with such general rings in this text. Therefore, to simplify terminology, from now on, by a “ring,” we shall always mean a commutative ring with unity.