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Chapter 4
Number Theory
We’ve now covered most of the basic techniques for writing proofs. So we’re going to start applying them to specific topics in mathematics, starting with number theory.
Number theory is a branch of mathematics concerned with the behavior of integers. It has very important applications in cryptography and in the design of randomized algorithms. Randomization has become an increasingly important technique for creating very fast algorithms for storing and retriev- ing objects (e.g. hash tables), testing whether two objects are the same (e.g. MP3’s), and the like. Much of the underlying theory depends on facts about which integers evenly divide one another and which integers are prime.
4.1 Factors and multiples
You’ve undoubtedly seen some of the basic ideas (e.g. divisibility) somewhat informally in earlier math classes. However, you may not be fully clear on what happens with special cases, e.g. zero, negative numbers. We also need clear formal definitions in order to write formal proofs. So, let’s start with
Definition: Suppose that a and b are integers. Then a divides b if b = an for some integer n. a is called a factor or divisor of b. b is called a multiple of a.
The shorthand for a divides b is a | b. Be careful about the order. The divisor is on the left and the multiple is on the right.
Some examples:
- 7 | 7 because 7 = 7 · 1
- 7 | 0 because 0 = 7 · 0, zero is divisible by any integer.
- 0 6 | 7 because 0 · n will always give you zero, never 7. Zero is a factor of only one number: zero.
- (−3) | 12 because 12 = 3 · − 4
- 3 | (−12) because −12 = 3 · − 4
An integer p is even exactly when 2 | p. The fact that zero is even is just a special case of the fact that zero is divisible by any integer.
4.2 Direct proof with divisibility
We can prove basic facts about divisibility in much the same way we proved basic facts about even and odd.
Claim 20 For any integers a,b,and c, if a | b and a | c then a | (b + c).
Proof: Let a,b,and c and suppose that a | b and a | c. Since a | b, there is an integer k such that b = ak (definition of divides). Similarly, since a | c, there is an integer j such that c = aj. Adding these two equations, we find that (b + c) = ak + aj = a(k + j). Since k and j are integers, so is k + j. Therefore, by the definition of divides, a | (b + c).
4.4 Prime numbers
We’re all familiar with prime numbers from high school. Firming up the details:
Definition: an integer q ≥ 2 is prime if the only positive factors of q are q and 1. An integer q ≥ 2 is composite if it is not prime.
For example, among the integers no bigger than 20, the primes are 2, 3, 5, 7, 11, 13, 17, and 19. Numbers smaller than 2 are neither prime nor composite.
A key fact about prime numbers is
Fundanmental Theorem of Arithmetic: Every integer ≥ 2 can be written as the product of one or more prime factors. Except for the order in which you write the factors, this prime factorization is unique.
The word “unique” here means that there is only one way to factor each integer.
For example, 260 = 2 · 2 · 5 · 13 and 180 = 2 · 2 · 3 · 3 · 5. We won’t prove this theorem right now, because it requires a proof tech- nique called “induction,” which we haven’t seen yet.
There are quite fast algorithms for testing whether a large integer is prime. However, even once you know a number is composite, algorithms for factoring the number are all fairly slow. The difficulty of factoring large composite numbers is the basis for a number of well-known cryptographic algorithms (e.g. the RSA algorithm).
4.5 GCD and LCM
If c divides both a and b, then c is called a common divisor of a and b. The largest such number is the greatest common divisor of a and b. Shorthand for this is gcd(a, b).
You can find the GCD of two numbers by inspecting their prime factor- izations and extracting the shared factors. For example, 140 = 2^2 · 5 · 7 and 650 = 2 · 52 · 13. So gcd(140, 6500) is 2 · 5 = 10.
Similarly, a common multiple of a and b is a number c such that a|c and b|c. The least common multiple (lcm) is the smallest positive number for which this is true. The lcm can be computed using the formula:
lcm(a, b) =
ab gcd(a, b)
For example, lcm(140, 650) = 14010 ·^650 = 9100. If two integers a and b share no common factors, then gcd(a, b) = 1. Such a pair of integers are called relatively prime.
If k is a non-zero integer, then k divides zero. the largest common divisor of k and zero is k. So gcd(k, 0) = gcd(0, k) = k. However, gcd(0, 0) isn’t defined. All integers are common divisors of 0 and 0, so there is no greatest one.
4.6 The division algorithm
The obvious way to compute the gcd of two integers is to factor both into primes and extract the shared factors. This is easy for small integers. How- ever, it quickly becomes very difficult for larger integers, both for humans and computers. Fortunately, there’s a fast method, called the Euclidean al- gorithm. Before presenting this algorithm, we need some background about integer division.
In general, when we divide one integer by another, we get a quotient and a remainder:
Theorem 1 (Division Algorithm) The Division Algorithm: For any in- tegers a and b, where b is positive, there are unique integers q (the quotient) and r (the remainder) such that a = bq + r and 0 ≤ r < b.
For example, if 13 is divided by 4, the quotient is 3 and the remainder is
gcd(a,b: positive integers) x := a y := b while (y > 0) begin r := remainder(x,y) x := y y := r end return x
Let’s trace this algorithm on inputs a = 105 and b = 252. Traces should summarize the values of the most important variables.
x y r = remainder(x, y) 105 252 105 252 105 42 105 42 21 42 21 0 21 0
Since x is smaller than y, the first iteration of the loop swaps x and y. After that, each iteration reduces the sizes of a and b, because a mod b is smaller than b. In the last iteration, y has gone to zero, so we output the value of x which is 21.
To verify that this algorithm is correct, we need to convince ourselves of two things. First, it must halt, because each iteration reduces the magnitude of y. Second, by our corollary above, the value of gcd(x, y) does not change from iteration to iteration. Moreover, gcd(x, 0) is x, for any non-zero integer x. So the final output will be the gcd of the two inputs a and b.
This is a genuinely very nice algorithm. Not only is it fast, but it involves very simple calculations that can be done by hand (without a calculator). It’s much easier than factoring both numbers into primes, especially as the individual prime factors get larger. Most of us can’t quickly see whether a large number is divisible by, say, 17.
4.8 Pseudocode
Notice that this algorithm is written in pseudocode. Pseudocode is an ab- stracted type of programming language, used to highlight the important structure of an algorithm and communicate between researchers who may not use the same programming language. It borrows many control constructs (e.g. the while loop) from imperative languages such as C. But details re- quired only for a mechanical compiler (e.g. type declarations for all variables) are omitted and equations or words are used to hide details that are easy to figure out.
If you have taken a programming course, pseudocode is typically easy to read. Many small details are not standardized, e.g. is the test for equality written = or ==? However, it’s usually easy for a human (though not a computer) to figure out what the author must have intended.
A common question is how much detail to use. Try to use about the same amount as in the examples shown in the notes. And think about how easily your pseudocode could be read by a classmate. Actual C or Java code is almost never acceptable pseudocode, because it is way too detailed.
4.9 A recursive version of gcd
We can also write gcd as a recursive algorithm
procedure gcd(a,b: positive integers) r := remainder(a,b) if (r = 0) return b else return gcd(b,r)
This code is very simple, because this algorithm has a natural recursive structure. Our corollary allows us to express the gcd of two numbers in terms of the gcd of a smaller pair of numbers. That is to say, it allows us to reduce a larger version of the task to a smaller version of the same task.
4.11 Proofs with congruence mod k
Let’s try using our definition to prove a simple fact about modular arithmetic:
Claim 24 For any integers a, b, c, d, and k, k positive, if a ≡ b (mod k) and c ≡ d (mod k), then a + c ≡ b + d (mod k).
Proof: Let a, b, c, d, and k be integers with k positive. Suppose that a ≡ b (mod k) and c ≡ d (mod k). Since a ≡ b (mod k), k | (a − b), by the definition of congruence mod k. Similarly, c ≡ d (mod k), k | (c − d). Since k | (a − b) and k | (c − d), we know by a lemma about divides (above) that k | (a − b) + (c − d). So k | (a + c) − (b + d) But then the definition of congruence mod k tells us that a + c ≡ b + d (mod k).
This proof can easily be modified to show that
Claim 25 For any integers a, b, c, d, and k, k positive, if a ≡ b (mod k) and c ≡ d (mod k), then ac ≡ bd (mod k).
So standard arithmetic operations interact well with our relaxed notion of equality.
4.12 Equivalence classes
The true power of modular conguence comes when we gather up a group of conguent integers and treat them all as a unit. Such a group is known as a congruence class or an equivalence class. Specifically, suppose that we fix a particular value for k. Then, if x is an integer, the equivalence class of x (written [x]) is the set of all integers congruent to x mod k. Or, equivalently, the set of integers that have remainder x when divided by k.
For example, suppose that we fix k to be 7. Then
[3] = { 3 , 10 , − 4 , 17 , − 11 ,.. .}
[1] = { 1 , 8 , − 6 , 15 , − 13 ,.. .}
[0] = { 0 , 7 , − 7 , 14 , − 14 ,.. .}
Notice that [−4], and [10] are exactly the same set as [3]. That is [−4] = [10] = [3]. So we have one object (the set) with many different names (one per integer in it). This is like a student apartment shared by Fred, Emily, Ali, and Michelle. The superficially different phrases “Emily’s apartment” and “Ali’s apartment” actually refer to one and the same apartment.
Having many names for the same object can become confusing, so people tend to choose a special preferred name for each object. For the k equiv- alence classes of integers mod k, mathematicians tend to prefer the names [0], [1],... , [k − 1]. Other names (e.g. [30] when k = 7) tend to occur only as intermediate results in calculations.
Because standard arithmetic operations interact well with modular con- gruence, we can set up a system of arithmetic on these equivalence classes. Specifically, we define addition and multiplication on equivalence classes by:
[x] + [y] = [x + y] [x] ∗ [y] = [x ∗ y]
So, (still setting k = 7) we can do computations such as
[4] + [10] = [4 + 10] = [14] = [0]
[−4] ∗ [10] = [− 4 ∗ 10] = [−40] = [2]
This new set of numbers ([0], [1],... , [k − 1]), with these modular rules of arithmetic and equality, is known as the “integers mod k” or Zk for short. For example, the addition and multiplication tables for Z 4 are:
- 4 [0] [1] [2] [3] [0] [0] [1] [2] [3] [1] [1] [2] [3] [0] [2] [2] [3] [0] [1] [3] [3] [0] [1] [2]
4.14 Variation in Terminology
In these notes, a divides b is defined to be true if b = an for some integer n. There is some variation among authors as to what happens when a is zero. Clearly, a non-zero number can’t be a multiple of zero. But is zero a multiple of itself? According to our definition, it is, but some authors explicitly exclude this special case. Fortunately, this is a special case that one rarely sees in practice. The greatest common divisor is also known as the highest common factor (HCF).
In the shorthand notation a ≡ b (mod k), the notation (mod k) is log- ically a modifier on our notion of equality (≡). In retrospect, it might have made more sense to write something like a ≡k b. However, a ≡ b (mod k) has become the standard notation and we have to live with it.
There are many variant notations for the quotient and remainder cre- ated by integer division, particularly if you include the functions built in to most programming languages. Popular names for the remainder include mod, modulo, rem, remainder, and %. The behavior on negative inputs dif- fers from language to language and, in some cases, from implementation to implementation.^2 This lack of standardization often results in hard-to-find program bugs.
Some authors use Z/nZ instead of Zn as the shorthand name for the inte- gers mod n. The notation Zn may then refer to a set which is structurally the same as the integers mod n, but in which multiplication and exponentiation are used as the basic operations, rather than addition and multiplication. The equivalence class of n is sometimes written n.
(^2) The “modulo operation” entry on wikipedia has a nice table of what happens in different languages.
