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Chapter 1
What Is Number Theory?
Number theory is the study of the set of positive whole numbers
1 , 2 , 3 , 4 , 5 , 6 , 7 ,... ,
which are often called the set of natural numbers. We will especially want to study the relationships between different sorts of numbers. Since ancient times, people have separated the natural numbers into a variety of different types. Here are some familiar and not-so-familiar examples:
odd 1 , 3 , 5 , 7 , 9 , 11 ,... even 2 , 4 , 6 , 8 , 10 ,... square 1 , 4 , 9 , 16 , 25 , 36 ,... cube 1 , 8 , 27 , 64 , 125 ,... prime 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 ,... composite 4 , 6 , 8 , 9 , 10 , 12 , 14 , 15 , 16 ,... 1 (modulo 4) 1 , 5 , 9 , 13 , 17 , 21 , 25 ,... 3 (modulo 4) 3 , 7 , 11 , 15 , 19 , 23 , 27 ,... triangular 1 , 3 , 6 , 10 , 15 , 21 ,... perfect 6 , 28 , 496 ,... Fibonacci 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 ,... Many of these types of numbers are undoubtedly already known to you. Oth- ers, such as the “modulo 4” numbers, may not be familiar. A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and sim- ilarly for the 3 (modulo 4) numbers. A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. The Fibonacci numbers are created by starting with 1 and 1. Then, to get the next number in the list, just add the previous two. Finally, a number is perfect if the sum of all its divisors, other than itself, adds back up to the
[Chap. 1] What Is Number Theory? 7
original number. Thus, the numbers dividing 6 are 1, 2, and 3, and 1 + 2 + 3 = 6. Similarly, the divisors of 28 are 1, 2, 4, 7, and 14, and
1 + 2 + 4 + 7 + 14 = 28.
We will encounter all these types of numbers, and many others, in our excursion through the Theory of Numbers.
Some Typical Number Theoretic Questions
The main goal of number theory is to discover interesting and unexpected rela- tionships between different sorts of numbers and to prove that these relationships are true. In this section we will describe a few typical number theoretic problems, some of which we will eventually solve, some of which have known solutions too difficult for us to include, and some of which remain unsolved to this day.
Sums of Squares I. Can the sum of two squares be a square? The answer is clearly “YES”; for example 32 + 4^2 = 5^2 and 52 + 12^2 = 13^2. These are examples of Pythagorean triples. We will describe all Pythagorean triples in Chapter 2.
Sums of Higher Powers. Can the sum of two cubes be a cube? Can the sum of two fourth powers be a fourth power? In general, can the sum of two nth^ powers be an nth^ power? The answer is “NO.” This famous problem, called Fermat’s Last Theorem, was first posed by Pierre de Fermat in the seventeenth century, but was not completely solved until 1994 by Andrew Wiles. Wiles’s proof uses sophisticated mathematical techniques that we will not be able to describe in detail, but in Chapter 30 we will prove that no fourth power is a sum of two fourth powers, and in Chapter 46 we will sketch some of the ideas that go into Wiles’s proof.
Infinitude of Primes. A prime number is a number p whose only factors are 1 and p.
- Are there infinitely many prime numbers?
- Are there infinitely many primes that are 1 modulo 4 numbers?
- Are there infinitely many primes that are 3 modulo 4 numbers?
The answer to all these questions is “YES.” We will prove these facts in Chapters 12 and 21 and also discuss a much more general result proved by Lejeune Dirichlet in 1837.
[Chap. 1] What Is Number Theory? 9
finitely many? To search for examples, the following formula is helpful:
1 + 2 + 3 + · · · + (n − 1) + n = n(n + 1) 2
There is an amusing anecdote associated with this formula. One day when the young Carl Friedrich Gauss (1777–1855) was in grade school, his teacher became so incensed with the class that he set them the task of adding up all the numbers from 1 to 100. As Gauss’s classmates dutifully began to add, Gauss walked up to the teacher and presented the answer, 5050. The story goes that the teacher was neither impressed nor amused, but there’s no record of what the next make-work assignment was!
There is an easy geometric way to verify Gauss’s formula, which may be the way he discovered it himself. The idea is to take two triangles consisting of 1 + 2 + · · · + n pebbles and fit them together with one additional diagonal of n + 1 pebbles. Figure 1.2 illustrates this idea for n = 6.
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z j j j j j j 1
j z j j j j j
j j z j j j j
j j j z j j j
j j j j z j j
j j j j j z j
j j j j j j z
(1 + 2 + 3 + 4 + 5 + 6) + 7 + (6 + 5 + 4 + 3 + 2 + 1) = 7^2
Figure 1.2: The Sum of the First n Integers
In the figure, we have marked the extra n + 1 = 7 pebbles on the diagonal with black dots. The resulting square has sides consisting of n + 1 pebbles, so in mathematical terms we obtain the formula 2(1 + 2 + 3 + · · · + n) + (n + 1) = (n + 1)^2 , two triangles + diagonal = square.
[Chap. 1] What Is Number Theory? 10
Now we can subtract n + 1 from each side and divide by 2 to get Gauss’s formula.
Twin Primes. In the list of primes it is sometimes true that consecutive odd num- bers are both prime. We have boxed these twin primes in the following list of primes less than 100:
3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 41 , 43 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97.
Are there infinitely many twin primes? That is, are there infinitely many prime numbers p such that p + 2 is also a prime? At present, no one knows the answer to this question.
Primes of the Form N 2 + 1. If we list the numbers of the form N 2 + 1 taking N = 1, 2 , 3 ,.. ., we find that some of them are prime. Of course, if N is odd, then N 2 + 1 is even, so it won’t be prime unless N = 1. So it’s really only interesting to take even values of N. We’ve highlighted the primes in the following list:
22 + 1 = 5 42 + 1 = 17 62 + 1 = 37 82 + 1 = 65 = 5 · 13 102 + 1 = 101 122 + 1 = 145 = 5 · 29 142 + 1 = 197 162 + 1 = 257 182 + 1 = 325 = 5^2 · 13 202 + 1 = 401.
It looks like there are quite a few prime values, but if you take larger values of N you will find that they become much rarer. So we ask whether there are infinitely many primes of the form N 2 + 1. Again, no one presently knows the answer to this question.
We have now seen some of the types of questions that are studied in the Theory of Numbers. How does one attempt to answer these questions? The answer is that Number Theory is partly experimental and partly theoretical. The experimental part normally comes first; it leads to questions and suggests ways to answer them. The theoretical part follows; in this part one tries to devise an argument that gives a conclusive answer to the questions. In summary, here are the steps to follow:
- Accumulate data, usually numerical, but sometimes more abstract in nature.
- Examine the data and try to find patterns and relationships.
- Formulate conjectures (i.e., guesses) that explain the patterns and relation- ships. These are frequently given by formulas.
[Chap. 1] What Is Number Theory? 12
1.6. For each of the following statements, fill in the blank with an easy-to-check crite- rion: (a) M is a triangular number if and only if is an odd square. (b) N is an odd square if and only if is a triangular number. (c) Prove that your criteria in (a) and (b) are correct.
Chapter 2
Pythagorean Triples
The Pythagorean Theorem, that “beloved” formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. In symbols,
a
b
c
a^2 + b^2 = c^2
Figure 2.1: A Pythagorean Triangle
Since we’re interested in number theory, that is, the theory of the natural num- bers, we will ask whether there are any Pythagorean triangles all of whose sides are natural numbers. There are many such triangles. The most famous has sides 3, 4, and 5. Here are the first few examples:
32 + 4^2 = 5^2 , 52 + 12^2 = 13^2 , 82 + 15^2 = 17^2 , 282 + 45^2 = 53^2.
The study of these Pythagorean triples began long before the time of Pythago- ras. There are Babylonian tablets that contain lists of parts of such triples, including quite large ones, indicating that the Babylonians probably had a systematic method for producing them. Even more amazing is the fact that the Babylonians may have
[Chap. 2] Pythagorean Triples 15
A primitive Pythagorean triple (or PPT for short) is a triple of num- bers (a, b, c) such that a, b, and c have no common factors^1 and satisfy a^2 + b^2 = c^2. Recall our checklist from Chapter 1. The first step is to accumulate some data. I used a computer to substitute in values for a and b and checked if a^2 + b^2 is a square. Here are some primitive Pythagorean triples that I found:
(3, 4 , 5), (5, 12 , 13), (8, 15 , 17), (7, 24 , 25), (20, 21 , 29), (9, 40 , 41), (12, 35 , 37), (11, 60 , 61), (28, 45 , 53), (33, 56 , 65), (16, 63 , 65).
A few conclusions can easily be drawn even from such a short list. For example, it certainly looks like one of a and b is odd and the other even. It also seems that c is always odd. It’s not hard to prove that these conjectures are correct. First, if a and b are both even, then c would also be even. This means that a, b, and c would have a common factor of 2, so the triple would not be primitive. Next, suppose that a and b are both odd, which means that c would have to be even. This means that there are numbers x, y, and z such that
a = 2x + 1, b = 2y + 1, and c = 2z.
We can substitute these into the equation a^2 + b^2 = c^2 to get
(2x + 1)^2 + (2y + 1)^2 = (2z)^2 , 4 x^2 + 4x + 4y^2 + 4y + 2 = 4z^2.
Now divide by 2,
2 x^2 + 2x + 2y^2 + 2y + 1 = 2z^2.
This last equation says that an odd number is equal to an even number, which is impossible, so a and b cannot both be odd. Since we’ve just checked that they cannot both be even and cannot both be odd, it must be true that one is even and
(^1) A common factor of a, b, and c is a number d such that each of a, b, and c is a multiple of d. For example, 3 is a common factor of 30, 42, and 105, since 30 = 3 · 10 , 42 = 3 · 14 , and 105 = 3 · 35 , and indeed it is their largest common factor. On the other hand, the numbers 10, 12, and 15 have no common factor (other than 1). Since our goal in this chapter is to explore some interesting and beautiful number theory without getting bogged down in formalities, we will use common factors and divisibility informally and trust our intuition. In Chapter 5 we will return to these questions and develop the theory of divisibility more carefully.
[Chap. 2] Pythagorean Triples 16
the other is odd. It’s then obvious from the equation a^2 + b^2 = c^2 that c is also odd. We can always switch a and b, so our problem now is to find all solutions in natural numbers to the equation
a^2 + b^2 = c^2 with
a odd, b even, a, b, c having no common factors.
The tools that we use are factorization and divisibility. Our first observation is that if (a, b, c) is a primitive Pythagorean triple, then we can factor a^2 = c^2 − b^2 = (c − b)(c + b).
Here are a few examples from the list given earlier, where note that we always take a to be odd and b to be even:
32 = 5^2 − 42 = (5 − 4)(5 + 4) = 1 · 9 , 152 = 17^2 − 82 = (17 − 8)(17 + 8) = 9 · 25 , 352 = 37^2 − 122 = (37 − 12)(37 + 12) = 25 · 49 , 332 = 65^2 − 562 = (65 − 56)(65 + 56) = 9 · 121.
It looks like c − b and c + b are themselves always squares. We check this obser- vation with a couple more examples:
212 = 29^2 − 202 = (29 − 20)(29 + 20) = 9 · 49 , 632 = 65^2 − 162 = (65 − 16)(65 + 16) = 49 · 81.
How can we prove that c − b and c + b are squares? Another observation ap- parent from our list of examples is that c − b and c + b seem to have no common factors. We can prove this last assertion as follows. Suppose that d is a common factor of c − b and c + b; that is, d divides both c − b and c + b. Then d also divides
(c + b) + (c − b) = 2c and (c + b) − (c − b) = 2b.
Thus, d divides 2 b and 2 c. But b and c have no common factor because we are assuming that (a, b, c) is a primitive Pythagorean triple. So d must equal 1 or 2. But d also divides (c − b)(c + b) = a^2 , and a is odd, so d must be 1. In other words, the only number dividing both c − b and c + b is 1, so c − b and c + b have no common factor.
[Chap. 2] Pythagorean Triples 18
For example, taking t = 1 in Theorem 2.1 gives a triple
s, s (^2) − 1 2 ,^
s^2 + 2
whose b and c entries differ by 1. This explains many of the examples that we listed. The following table gives all possible triples with s ≤ 9.
s t a = st b = s^2 − t^2 2
c = s^2 + t^2 2 3 1 3 4 5 5 1 5 12 13 7 1 7 24 25 9 1 9 40 41 5 3 15 8 17 7 3 21 20 29 7 5 35 12 37 9 5 45 28 53 9 7 63 16 65
A Notational Interlude
Mathematicians have created certain standard notations as a shorthand for various quantities. We will keep our use of such notation to a minimum, but there are a few symbols that are so commonly used and are so useful that it is worthwhile to introduce them here. They are
N = the set of natural numbers = 1, 2 , 3 , 4 ,... , Z = the set of integers =... − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 ,... , Q = the set of rational numbers (i.e., fractions).
In addition, mathematicians often use R to denote the real numbers and C for the complex numbers, but we will not need these. Why were these letters chosen? The choice of N, R, and C needs no explanation. The letter Z for the set of inte- gers comes from the German word “Zahlen,” which means numbers. Similarly, Q comes from the German “Quotient” (which is the same as the English word). We will also use the standard mathematical symbol ∈ to mean “is an element of the set.” So, for example, a ∈ N means that a is a natural number, and x ∈ Q means that x is a rational number.
Exercises
2.1. (a) We showed that in any primitive Pythagorean triple (a, b, c), either a or b is even. Use the same sort of argument to show that either a or b must be a multiple of 3.
[Chap. 2] Pythagorean Triples 19
(b) By examining the above list of primitive Pythagorean triples, make a guess about when a, b, or c is a multiple of 5. Try to show that your guess is correct.
2.2. A nonzero integer d is said to divide an integer m if m = dk for some number k. Show that if d divides both m and n, then d also divides m − n and m + n.
2.3. For each of the following questions, begin by compiling some data; next examine the data and formulate a conjecture; and finally try to prove that your conjecture is correct. (But don’t worry if you can’t solve every part of this problem; some parts are quite difficult.) (a) Which odd numbers a can appear in a primitive Pythagorean triple (a, b, c)? (b) Which even numbers b can appear in a primitive Pythagorean triple (a, b, c)? (c) Which numbers c can appear in a primitive Pythagorean triple (a, b, c)?
2.4. In our list of examples are the two primitive Pythagorean triples
332 + 56^2 = 65^2 and 162 + 63^2 = 65^2.
Find at least one more example of two primitive Pythagorean triples with the same value of c. Can you find three primitive Pythagorean triples with the same c? Can you find more than three?
2.5. In Chapter 1 we saw that the nth^ triangular number Tn is given by the formula
Tn = 1 + 2 + 3 + · · · + n = n(n + 1) 2 .
The first few triangular numbers are 1 , 3 , 6 , and 10. In the list of the first few Pythagorean triples (a, b, c), we find (3, 4 , 5), (5, 12 , 13), (7, 24 , 25), and (9, 40 , 41). Notice that in each case, the value of b is four times a triangular number. (a) Find a primitive Pythagorean triple (a, b, c) with b = 4T 5. Do the same for b = 4T 6 and for b = 4T 7. (b) Do you think that for every triangular number Tn, there is a primitive Pythagorean triple (a, b, c) with b = 4Tn? If you believe that this is true, then prove it. Otherwise, find some triangular number for which it is not true.
2.6. If you look at the table of primitive Pythagorean triples in this chapter, you will see many triples in which c is 2 greater than a. For example, the triples (3, 4 , 5), (15, 8 , 17), (35, 12 , 37), and (63, 16 , 65) all have this property. (a) Find two more primitive Pythagorean triples (a, b, c) having c = a + 2. (b) Find a primitive Pythagorean triple (a, b, c) having c = a + 2 and c > 1000. (c) Try to find a formula that describes all primitive Pythagorean triples (a, b, c) having c = a + 2.
2.7. For each primitive Pythagorean triple (a, b, c) in the table in this chapter, compute the quantity 2 c − 2 a. Do these values seem to have some special form? Try to prove that your observation is true for all primitive Pythagorean triples.
2.8. Let m and n be numbers that differ by 2 , and write the sum (^) m^1 + (^) n^1 as a fraction in lowest terms. For example, 12 + 14 = 34 and 13 + 15 = 158.
Chapter 3
Pythagorean Triples
and the Unit Circle
In the previous chapter we described all solutions to
a^2 + b^2 = c^2
in whole numbers a, b, c. If we divide this equation by c^2 , we obtain
( (^) a c
b c
So the pair of rational numbers (a/c, b/c) is a solution to the equation
x^2 + y^2 = 1.
Everyone knows what the equation x^2 + y^2 = 1 looks like: It is a circle C of radius 1 with center at (0, 0). We are going to use the geometry of the circle C to find all the points on C whose xy-coordinates are rational numbers. Notice that the circle has four obvious points with rational coordinates, (± 1 , 0) and (0, ±1). Suppose that we take any (rational) number m and look at the line L going through the point (− 1 , 0) and having slope m. (See Figure 3.1.) The line L is given by the equation L : y = m(x + 1) (point–slope formula).
It is clear from the picture that the intersection C ∩L consists of exactly two points, and one of those points is (− 1 , 0). We want to find the other one. To find the intersection of C and L, we need to solve the equations
x^2 + y^2 = 1 and y = m(x + 1)
[Chap. 3] Pythagorean Triples and the Unit Circle 22
C
L = line with slope m (–1,0 )
Figure 3.1: The Intersection of a Circle and a Line
for x and y. Substituting the second equation into the first and simplifying, we need to solve
x^2 +
m(x + 1)
x^2 + m^2 (x^2 + 2x + 1) = 1 (m^2 + 1)x^2 + 2m^2 x + (m^2 − 1) = 0.
This is just a quadratic equation, so we could use the quadratic formula to solve for x. But there is a much easier way to find the solution. We know that x = − 1 must be a solution, since the point (− 1 , 0) is on both C and L. This means that we can divide the quadratic polynomial by x + 1 to find the other root:
(m^2 + 1)x + (m^2 − 1) x + 1
(m^2 + 1)x^2 + 2m^2 x + (m^2 − 1).
So the other root is the solution of (m^2 + 1)x + (m^2 − 1) = 0, which means that
x = 1 − m^2 1 + m^2
Then we substitute this value of x into the equation y = m(x + 1) of the line L to find the y-coordinate,
y = m(x + 1) = m
1 − m^2 1 + m^2
2 m 1 + m^2
Thus, for every rational number m we get a solution in rational numbers ( 1 − m^2 1 + m^2
2 m 1 + m^2
to the equation x^2 + y^2 = 1.
[Chap. 3] Pythagorean Triples and the Unit Circle 24
(a) If u and v have a common factor, explain why (a, b, c) will not be a primitive Pytha- gorean triple. (b) Find an example of integers u > v > 0 that do not have a common factor, yet the Pythagorean triple (u^2 − v^2 , 2 uv, u^2 + v^2 ) is not primitive. (c) Make a table of the Pythagorean triples that arise when you substitute in all values of u and v with 1 ≤ v < u ≤ 10. (d) Using your table from (c), find some simple conditions on u and v that ensure that the Pythagorean triple (u^2 − v^2 , 2 uv, u^2 + v^2 ) is primitive. (e) Prove that your conditions in (d) really work.
3.2. (a) Use the lines through the point (1, 1) to describe all the points on the circle
x^2 + y^2 = 2 whose coordinates are rational numbers. (b) What goes wrong if you try to apply the same procedure to find all the points on the circle x^2 + y^2 = 3 with rational coordinates?
3.3. Find a formula for all the points on the hyperbola
x^2 − y^2 = 1
whose coordinates are rational numbers. [Hint. Take the line through the point (− 1 , 0) having rational slope m and find a formula in terms of m for the second point where the line intersects the hyperbola.]
3.4. The curve y^2 = x^3 + 8
contains the points (1, −3) and (− 7 / 4 , 13 /8). The line through these two points intersects the curve in exactly one other point. Find this third point. Can you explain why the coordinates of this third point are rational numbers?
3.5. Numbers that are both square and triangular numbers were introduced in Chapter 1, and you studied them in Exercise 1.1. (a) Show that every square–triangular number can be described using the solutions in positive integers to the equation x^2 − 2 y^2 = 1. [Hint. Rearrange the equation m^2 = 1 2 (n (^2) + n).] (b) The curve x^2 − 2 y^2 = 1 includes the point (1, 0). Let L be the line through (1, 0) having slope m. Find the other point where L intersects the curve. (c) Suppose that you take m to equal m = v/u, where (u, v) is a solution to u^2 − 2 v^2 =
- Show that the other point that you found in (b) has integer coordinates. Further, changing the signs of the coordinates if necessary, show that you get a solution to x^2 − 2 y^2 = 1 in positive integers. (d) Starting with the solution (3, 2) to x^2 − 2 y^2 = 1, apply (b) and (c) repeatedly to find several more solutions to x^2 − 2 y^2 = 1. Then use those solutions to find additional examples of square–triangular numbers.
[Chap. 3] Pythagorean Triples and the Unit Circle 25
(e) Prove that this procedure leads to infinitely many different square-triangular numbers. (f) Prove that every square–triangular number can be constructed in this way. (This part is very difficult. Don’t worry if you can’t solve it.)
